Mediterranean Borderlands: Early English Merchants in the Levant

arXiv:1110.6092v4 [math.GR] 24 Jan 2016 ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Abstract. Consider an arithmetic group G(OS ), where G is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers OS of a number field K with respect to a finite set of places S. For each n ∈ N, let Rn (G(OS )) denote the number of irreducible complex representations of G(OS ) of dimension at most n. The degree of representation growth α(G(OS )) = limn→∞ log Rn (G(OS ))/ log n is finite if and only if G(OS ) has the weak Congruence Subgroup Property. We establish that for every G(OS ) with the weak Congruence Subgroup Property the invariant α(G(OS )) is already determined by the absolute root system of G. To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions K ⊂ L. We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky’s conjecture to Serre’s conjecture on the weak Congruence Subgroup Property, which it refines. Contents 1. Introduction and Main Results 1.1. Background and Motivation 1.2. Discussion of Main Results 2. Reduction of the Main Results to Theorem 2.8 3. Base Change for Finite Groups of Lie Type 3.1. Finite Groups of Lie Type 3.2. Applications to Finite Quotients of Arithmetic Groups 4. Relative Zeta Functions, Kirillov Orbit Method and Model Theoretic Background 4.1. Relative Zeta Functions and Cohomology 4.2. Kirillov Orbit Method 4.3. Quantifier-Free Definable Sets and Functions 4.4. Valued Fields 5. Parametrizing Representations 5.1. Relative Orbit Method 5.2. The Stabilizer of ΞR 5.3. The Lie Algebra Associated to the Stabilizer of ΞR 2 2 3 7 14 14 26 27 27 29 32 38 40 40 43 48 2 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL 5.4. Proof of Theorem 5.1 6. Quantifier-Free Integrals 6.1. Uniform Formulae for Quantifier-Free Integrals 6.2. Proof of Theorem 6.2 6.3. Proof of Theorem 2.8 References 51 56 56 57 63 64 1. Introduction and Main Results 1.1. Background and Motivation. One of the aims of this paper is to prove a variant of a conjecture of Larsen and Lubotzky on the representation growth of irreducible lattices in higher rank semi-simple groups. We recall that the representation growth of an arbitrary group G is given by the asymptotic behavior of the sequence Rn (G), n ∈ N, where Rn (G) denotes the number of equivalence classes of irreducible complex representations of G of dimension at most n. Whenever G is a topological (resp. algebraic) group, we restrict the investigation without further comment to continuous (resp. rational) representations. According to Margulis’ Arithmeticity Theorem, the lattices in question arise in the following way. Consider an arithmetic group G(OS ), where OS is the ring of S-integers of a number field K with respect to a finite set of places S of K and G is an affine group scheme over OS whose generic fiber is connected, simplyP connected absolutely almost simple. Suppose further that the S-rank of G, i.e., rkS G = v∈S rkKv G, is at least 2 and that an infinite place v is included in S if rkKv G ≥ 1. A theorem of Borel and Harish–Chandra shows that the image of G(OS ) under the diagonal Q embedding is indeed an irreducible lattice in the higher rank semi-simple group H = v∈S G(Kv ). Moreover, Margulis proved that this construction produces, up to commensurability, essentially all irreducible lattices in higher rank semi-simple groups. Precise notions and a more complete description can be found in [33]. In this paper, we call a group arithmetic if it is commensurable to a group of the form G(OS ) as above. In particular, all arithmetic groups that we consider are defined in characteristic 0. The study of representation growth for arithmetic groups was initiated by Lubotzky and Martin in [30]. They showed that, whenever Γ is commensurable to G(OS ) as above and rkKp G ≥ 1 for every finite place p ∈ S, then the growth of the sequence Rn (Γ), n ∈ N, is bounded polynomially in n if and only if G(OS ) has the weak Congruence Subgroup \ c Property. To discuss the latter, let G(O S ) and OS denote the profinite completions of the group G(OS ) and the ring OS . Furthermore, we write Op for the completion of the ring of integers O of K at a prime p. The group G(OS ) has the weak Congruence Subgroup Property (wCSP) if the kernel of the natural map Y \ c ∼ G(Op ), G(O S ) −→ G(OS ) = p∈Spec(O)rS ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 3 is finite. A long-standing conjecture of Serre asserts, in particular, that G(OS ) has the wCSP whenever rkS G ≥ 2 and rkKp G ≥ 1 for every finite place p ∈ S. This part of Serre’s conjecture is known to be true in many cases; e.g., it holds for groups yielding non-uniform irreducible lattices in higher rank semi-simple groups. For more information see [37, Chapter 9.5], [39] or [38], and the references therein. Next we recall the definition of the representation zeta function of a group G and its abscissa of convergence, which captures the degree of representation growth of G. Definition. Let G be a group such that Rn (G) is finite for all n ∈ N. The representation zeta function of G is the Dirichlet generating series X ζG (s) = (dim ̺)−s , ̺∈Irr(G) where Irr(G) is the set of equivalence classes of finite-dimensional irreducible complex representations ̺ of G and s ∈ C is a complex variable. The abscissa of convergence of ζG (s) is the infimum of all σ ∈ R such that the series ζG (s) converges absolutely for all s ∈ C with Re(s) > σ; we denote this invariant by α(G). In particular, α(G) = ∞ if ζG (s) diverges for all s ∈ C. Whenever a group G, as in the definition above, possesses infinitely many finitedimensional irreducible complex representations, the abscissa of convergence α(G) is related to the asymptotic behavior of the sequence Rn (G) by the equation log Rn (G) (1.1) α(G) = lim sup . log n n→∞ In the case of an arithmetic group Γ = G(OS ), as described above, Lubotzky and Martin’s result in [30] can therefore be stated as follows: Γ has the wCSP if and only if α(Γ) < ∞. In this sense the invariant α(Γ) provides a means to study the wCSP in a quantitative Rn (Γ) way. We also remark that if Γ has the wCSP, then α(Γ) = limn→∞ loglog is actually a n α(Γ)+o(1) limit, hence Rn (Γ) = n ; this is shown implicitly in [2]. 1.2. Discussion of Main Results. In this paper we establish new quantitative results regarding the representation growth of arithmetic groups with the wCSP. Our first main theorem is the following. Theorem 1.1. Let Φ be an irreducible root system. Then there exists a constant αΦ such that, for every arithmetic group G(OS ), where OS is the ring of S-integers of a number field K with respect to a finite set of places S and G is an affine group scheme over OS whose generic fiber is connected, simply connected absolutely almost simple with absolute root system Φ, the following holds: if G(OS ) has the wCSP, then α(G(OS )) = αΦ . The theorem highlights two challenging open problems, namely to determine the constants αΦ and to establish finer asymptotics for the representation growth of arithmetic groups with the wCSP. Even at the conjectural level we are presently very far from solving these problems. The main theorem in [2] shows that αΦ ∈ Q for all Φ. Furthermore, 4 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL 1 (see [28, Theorem 8.1]) and αAℓ ≤ 22 for all ℓ ∈ N, with similar bounds on αΦ ≥ 15 other root systems (see [1]). The only precisely known values are αA1 = 2 (see [28, Theorem 10.1]) and αA2 = 1 (see [3, Theorem C]). In fact, for arithmetic groups Γ = G(OS ) with the wCSP which arise from affine group schemes G of type A1 or A2 , even finer asymptotics of the representation growth of Γ have been established. If G has absolute root system A1 , then ζΓ (s) admits a meromorphic continuation beyond its abscissa of convergence and has a simple pole at s = 2 (compare [28] and [4]); consequently, Rn (Γ) = (cΓ + o(1))n2 for a constant cΓ ∈ R. Similarly, if G has absolute root system A2 , then ζΓ (s) has a meromorphic continuation beyond its abscissa of convergence and a double pole at s = 1 (see [4]); consequently, Rn (Γ) = (cΓ + o(1))n log n for a constant cΓ ∈ R. For general Γ, it remains open whether and how far ζΓ (s) can be extended meromorphically and, if so, whether the order of the resulting pole at s = α(Γ) depends only on the absolute root system Φ. Besides its intrinsic group theoretic importance, the invariant α(G(OS )) of an arithmetic group G(OS ) is also related to the singularities of deformation varieties of surface groups inside G(C); see [1]. Furthermore it is significant for the volumes of the moduli space of U-local systems on algebraic curves, where U is a compact group. We refer to Witten’s paper [43] for the case where U is a compact Lie group and to [1] for the case where U is a maximal compact subgroup in the adelic group G(AK ). Finally, we remark that Theorem 1.1 is similar in spirit to the main result of [31], which pins down the subgroup growth of irreducible lattices in higher rank semi-simple groups (modulo Serre’s conjecture and the generalized Riemann Hypothesis). We emphasize that the subgroup growth of such lattices is always faster than polynomial and, in fact, our proofs and methods are completely different from those used in [31]. Let us now focus on the representation growth of irreducible lattices in semi-simple locallyQ compact groups. Let H be a semi-simple group in characteristic 0 of the form H = rj=1 Hj (Fj ), where each Fj is a local field of characteristic 0 and each Hj is a connected, almost simple Fj -group. As indicated at the end of Section 1.1, for every arithmetic irreducible lattice Γ in H we may regard α(Γ) as a quantitative measure for the wCSP. Moreover, Serre’s conjecture on the Congruence Subgroup Problem asserts that the question whether such a lattice Γ has the wCSP, equivalently whether α(Γ) < ∞, does not depend on the particular choice of Γ, but is controlled byPthe ambient group H. Larsen and Lubotzky conjectured that, if H has higher rank, i.e., rj=1 rkFj Hj ≥ 2, then the abscissa of convergence α(Γ) is the same for all irreducible lattices Γ in H; see [28, Conjecture 1.5]. We establish the following variant. Theorem 1.2. Let H be a semi-simple group in characteristic 0, and let Γ1 , Γ2 be two arithmetic irreducible lattices in H, both having the wCSP or, equivalently, satisfying α(Γ1 ), α(Γ2 ) < ∞. Then α(Γ1 ) = α(Γ2 ). This unconditional result and Margulis’ Arithmeticity Theorem immediately reduce Larsen and Lubotzky’s conjecture to the original conjecture of Serre. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 5 Theorem 1.3. Let H be a higher-rank semi-simple group in characteristic 0. Assuming Serre’s conjecture, for any two irreducible lattices Γ1 , Γ2 in H we have α(Γ1 ) = α(Γ2 ). We emphasize that our results are at the same time weaker and stronger than [28, Conjecture 1.5]. They are weaker, since we do not prove Serre’s conjecture, and stronger, since Theorem 1.1 shows that the abscissa of convergence depends only on the absolute root system associated to the ambient semi-simple group H. Central to this paper are new insights into the behavior of the abscissa of convergence under base change. More precisely, we consider the relation between the abscissae of convergence for groups G(O1 ) and G(O2 ), where O1 ⊂ O2 is a ring extension. We initiated this study in [3] in the local case, where each Oi is a compact discrete valuation ring of characteristic 0. In the present paper we consider the global case, where each Oi is the ring of Si -integers in a number field Ki . It is convenient to organize the results on base change of arithmetic groups into two theorems. Theorem 1.4 links the abscissa of convergence for an arithmetic group G(OS ) b over the integral adeles with the wCSP to the abscissa of convergence for the group G(O) Q b = O p∈Spec(O) Op . Key to this are results of Larsen and Lubotzky, such as an Euler product factorization for representation zeta functions of arithmetic groups and results on the representation growth of Lie groups, and [3, Theorem B]. The latter is a base change result in the local case; we record a relevant corollary as Theorem 2.2 below. Theorem 1.4. Let K be a number field with ring of integers O and let S be a finite set of places of K. Let G be an affine group scheme defined over OS whose generic fiber is connected and simply connected semi-simple. Suppose that G(OS ) has the wCSP. Then
b . α(G(OS )) = α G(O) Theorem 1.5 incorporates the main thrust of the current paper. It relates the abscissae cK ) and G(O cL ), where K ⊂ L is an extension of of convergence for the adelic groups G(O number fields. Theorem 1.5. Let K ⊂ L be number fields with rings of integers OK ⊂ OL , and let G be an affine group scheme defined over
OK whose generic fiber is connected and simply
c c connected semi-simple. Then α G(OK ) = α G(OL ) . It is noteworthy that the global situation is more rigid than the local one: in the local case, the abscissa of convergence is monotone non-decreasing with respect to base change, but it can be strictly increasing; see Remark 2.3. At a formal level we can summarize the base change theorems as follows. Theorem 1.4 means that α(G(O1 )) = α(G(O2)) when Spec(O2 ) → Spec(O1) is an open embedding. Theorem 1.5 means that α(G(O1 )) = α(G(O2 )) when Spec(O2 ) → Spec(O1 ) is finite. Taken together, they mean that α(G(O1)) = α(G(O2 )) when Spec(O2 ) → Spec(O1 ) has finite fibers. Theorem 1.5 is more difficult to prove than its local counterpart Theorem 2.2. Our approach is based on close approximations of the representation zeta functions associated 6 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL to groups of the form G(Op ). A central feature of these approximations is that they are uniform, as O ranges over the set of all finite extensions of a fixed global ring and p ranges over a cofinite set of primes of O. We give such approximations in Theorem 2.8. Similar approximations for zeta functions of the finite groups of Lie type G(O/p) are derived in Theorem 3.1, using Deligne–Lusztig theory. This allows us to control representations of Q b that factor through G(O) p G(O/p), but does not account for other representations. In contrast to the situation for finite groups of Lie type, the representation theory of groups over local rings is at present poorly understood. Instead of enumerating representations directly, we follow in the proof of Theorem 2.8 Weil’s idea to express local factors of zeta functions as p-adic integrals. We show that in our case the local factors can be approximated by a class of p-adic integrals involving quantifier-free definable functions (Theorem 5.1) and that such integrals admit uniform formulae (Theorem 6.2). In this context we employ tools from the model theory of valued fields, such as partial elimination of quantifiers in the theory of Henselian valued fields of residue characteristic 0. Working with quantifier-free definable functions, we strike a balance between being able to approximate the relevant local factors in the first place and being able to derive uniform formulae for the resulting integrals. At present it is unknown whether one may use more elementary classes of functions, such as polynomials, to carry out an equally effective approximation. Organization. In Section 2 we prove Theorem 1.4, state Theorem 2.8, and prove Theorems 1.5, 1.1, and 1.2, using Theorem 2.8. The rest of the paper is dedicated to proving Theorem 2.8. In Section 3 we prove Theorem 3.1, which is a variant of Theorem 2.8 for zeta functions of semi-simple algebraic groups over finite fields, and apply it to finite quotients of arithmetic groups. In Section 4 we collect some results about relative representation zeta functions, the Kirillov orbit method, and model theory. In Section 5 we prove that the local factors of representation zeta functions of arithmetic groups are approximated by integrals of quantifier-free definable functions. In Section 6 we prove that integrals of quantifier-free definable functions have a uniform formula, and finish the proof of Theorem 2.8. Notation. All affine group schemes appearing in this paper are algebraic. For reference purposes we summarize some of the notation used frequently. ◦ G, H denote affine group schemes; in this context g refers to the Lie algebra of G, but sometimes g denotes a more general Lie lattice. ◦ Φ denotes a root system; rk Φ its rank and Φ+ a choice of positive roots; C(G, T, E) and C(Φ) are defined in (3.6), (3.7). ◦ Γ, ∆ denote arithmetic groups. ◦ K, L denote number fields, with rings of integers OK , OL . ◦ p denotes a prime of OK and q a prime of OL . ◦ Kp and OK,p = OKp denote the completions of K and OK at p; similarly, Lq and OL,q = OLq are the completions of L and OL at q. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 7 ◦ A is a semi-ring with an ideal A+ and ξa,q a Dirichlet polynomial; see Definition 2.6. ◦ GrLie (g) and Grnilp Lie (g) are the Grassmannians of Lie subalgebras and of nilpotent Lie subalgebras of a Lie algebra g; see Definition 4.11. ◦ F, k, Γ are the sorts of the Denef–Pas language of valued fields; see Section 4.4. ◦ T fields , T fields,R , T perf.-fields,p,R , T Hen,0 , and T Hen,K,0 denote the first-order theories of certain types of fields; see Sections 4.3 and 4.4. ◦ Π, Ξ and decorations thereof refer to relative orbit method functions; see Section 5.1. ◦ X , Y are quantifier-free definable sets introduced in Definitions 5.2 and 5.12. ◦ R, L, S and decorations thereof refer to definable functions/families of Lie algebras/groups over X and Y ; see Section 5.1 and also Theorem 5.18. Additional summaries of more specialised notation can be found in Sections 5.1 and 5.2. Acknowledgments. Avni was supported by NSF Grant DMS-0901638. Onn was supported by ISF grant 382/11. We acknowledge support from the EPSRC. Many thanks to Udi Hrushovski, Michael Larsen, and Alex Lubotzky for lots of helpful conversations and the referees for a number of comments that improved the exposition of the paper. 2. Reduction of the Main Results to Theorem 2.8 In this section we prove Theorems 1.1, 1.2, and 1.5, modulo Theorem 2.8 stated below. For this purpose we fix the following notation. ◦ K is a number field, O = OK its ring of integers, and OS the ring of S-integers for a finite set of places S. ◦ G is an affine group scheme defined over OS whose generic fiber is connected and simply connected semi-simple. ◦ Γ = G(OS ) has the wCSP. The groups that we consider in this article have the property that their categories of finite-dimensional complex representations are semi-simple. For example, Γ satisfies this condition since, for every finite-dimensional complex representation ̺ of Γ, the Zariski closure of ̺(Γ) is a reductive algebraic group in characteristic 0. Our starting point is an Euler product factorization for the representation zeta function of a suitable subgroup ∆ ⊂ Γ. The following elementary lemma will be used repeatedly. Lemma 2.1. Let G1 and G2 be groups such that their categories of finite-dimensional complex representations are semi-simple. If Rn (G1 ) and Rn (G2 ) are finite for all n ∈ N, then ζG1 ×G2 = ζG1 ζG2 . In particular, α(G1 × G2 ) = max{α(G1 ), α(G2)}. Proof. This follows from the fact that the irreducible complex representations of G1 × G2 are the tensor products of irreducible complex representations of G1 and G2 .  8 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Considering pro-algebraic completions, one finds that there is a finite-index subgroup ∆ ⊂ Γ such that Y [K:Q] (2.1) ζ∆ = ζG(C) · ζ ∆p , p∈S / where the product ranges over the primes p ∈ Spec(O) r S = Spec(OS ) and ∆p is the closure, in the p-adic topology, of the image of ∆ under the embedding Γ → G(Op); see [28, Theorem 3.3 and Proposition 4.6]. Furthermore, the generating series ζG(C) counts only rational representations, and the generating series ζ∆p count only continuous representations. Since ∆ has finite index in Γ, the Strong Approximation Theorem implies that ∆p is open in G(Op ), for every p, and ∆p = G(Op ), for all but finitely many primes. It is well-known that the abscissa of convergence for groups is a commensurability invariant (see [30, Lemma 2.2]; we prove a more general version in Lemma 2.11). In particular, α(Γ) is equal to α(∆). By [28, Theorem 5.1] and [28, Proposition 6.6], we have α(G(C)) ≤ α(G(Op)), for every p ∈ Spec(OS ). Q Therefore, (2.1) shows that α(Γ) is equal to the abscissa of convergence of the product p∈S of the / ζ∆p . By another application Q commensurability invariance, α(Γ) is equal to the abscissa of convergence of p∈S / ζG(Op ) . To deduce Theorem 1.4 we need to justify that the abscissa of convergence of the Q product p ζG(Op ) , ranging over all primes p ∈ Spec(O), is unchanged by omitting finitely many factors. This is a consequence of the following more general result. Theorem 2.2. Let K be a number field with ring of integers OK , and let G be an affine group scheme over OK whose generic fiber is connected and simply connected semi-simple. Then, for every finite extension L of K with ring of integers OL and every prime q of OL , there are infinitely many primes p of OK such that α(G(OL,q)) ≤ α(G(OK,p)). Theorem 2.2 is a corollary of [3, Theorem B], whose proof uses p-adic integrals to analyze the representation zeta functions of q-adic groups such as G(OL,q ). The connection to p-adic integrals, and more generally definable integrals in the sense of model theory, is [3, Corollary 3.7] (see also [24, Lemma 4.1]). The notion of a quantifier-free definable function is explained in Section 4. It follows from [3, Corollary 3.7] that there are d ∈ N and quantifier-free definable functions ϕ1 , ϕ2 such that, for every finite extension L of K, every prime q of OL , and every sufficiently large integer r, the representation zeta function of the rth principal congruence subgroup G(r) (OL,q ) = ker(G(OL,q ) → G(OL /qr )) can be expressed as follows: Z r·dim G |ϕ1 (x)|q |ϕ2 (x)|−s (2.2) ζG(r) (OL,q ) (s) = |OL /q| q dλ(x), d OL,q where the absolute value in the integrand is the q-adic one, and λ is the additive Haar d measure on Ldq normalized so that λ(OL,q ) = 1. Since the abscissae of convergence for G(OL,q ) and its rth principal congruence subgroup are equal, the claim in Theorem 2.2 may thus be reduced to a similar claim for integrals of the form (2.2). The main point is ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 9 that the functions ϕ1 , ϕ2 are independent of L and q, allowing us to compare the integrals for varying fields and primes. Remark 2.3. Theorem 2.2 can be regarded as a local analog of Theorem 1.5. We explain why a more naive analog is false. Suppose that K ⊂ L is a finite extension of number fields and that q is a prime of OL lying over a prime p of OK . Then [3, Theorem B] implies that α(G(OK,p)) ≤ α(G(OL,q)). However, in contrast to the global case considered in Theorem 1.5, the inequality can be strict. For example, let D be a central division algebra of degree d over a non-archimedean local field F , and let G be an affine group scheme over OF such that G(F ) is the group of norm 1 elements in D. Then the abscissa of convergence for G(OF ) is 2/d; see [28, Theorem 7.1]. But if F ⊂ E is an extension such that D splits over E, then α(G(OE )) ≥ 1/15 by [28, Theorem 8.1], and hence strictly greater than 2/d for d > 30. We move on to the proof of Theorem 1.5. The description in (2.2) of representation zeta functions of principal congruence subgroups G(r) (Op ) as p-adic integrals is of limited b use for determining α(G(O)). This is due to the fact that the infinite product of such Q b = congruence subgroups is not of finite index in G(O) p G(Op ). Facing the challenge to deal with the groups G(Op ), and not merely congruence subgroups of sufficiently large index, we approximate their representation zeta functions in the following sense. P P −s −s be Dirichlet generating and g(s) = ∞ Definition 2.4. Let f (s) = ∞ n=1 bn n n=1 an n series, i.e., Dirichlet series with integer coefficients an , bn ≥ 0. Let C ∈ R. Suppose that σ0 ∈ R≥0 is greater than or equal to the abscissae of convergence of f and g. We write f .C g for σ > σ0 if f (σ) ≤ C g(σ) for every σ ∈ R with σ > σ0 . We write f .C g, without specifying the domain, if f and g have the same abscissa of convergence α and if f .C g for σ > max{0, α}. Finally, we write f ∼C g if f .C g and g .C f . 1+σ We routinely use the fact that f .C1 g and g .C2 h imply f .C1 C2 h. Lemma 2.5. Let f,Qg be Dirichlet generating Q series with abscissae of convergence αf , αg . ∞ (1 + f ) and g = Suppose that f = ∞ m m=1 (1 + gm ), where fm , gm are Dirichlet m=1 generating series with vanishing constant terms, and, for every m ∈ N, let βm denote the abscissa of convergence of gm . Suppose further that, for each ε > 0, there is C(ε) ∈ R>0 such that, for all m, fm .C(ε) gm for σ > βm + ε. Then αf ≤ αg . Proof. The assumptions imply that the abscissa of convergence of fm is less than or equal to βm for every m. The abscissa of convergence of a Dirichlet generating series is determined by its behavior on the real axis. Fix ε > 0, Q and let σ ∈ R with σ > max{0, αg } + ε, so that σ > βm + ε for all m. Then g(σ) = m (1 + gm (σ)) P and hence P 1+σ g (σ) converge. As f (σ) ≤ C(ε) g (σ) for all m, this implies that m m m m m fm (σ) Q and hence f (σ) = m (1 + fm (σ)) converge. Thus σ > αf . Letting ε tend to 0, we deduce that αg ≥ αf .  10 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL We now introduce a semi-ring A whose elements a are used to index Dirichlet polynomials ξa,q approximating certain Dirichlet generating series. Definition 2.6. Let A be the collection of all finite subsets a ⊂ Z≥0 × Z>0 ∪ {(0, 0)}. We turn A into a commutative unital semi-ring by defining the sum of a, b ∈ A as a+b = a∪b and their product as a · b = {u + v | u ∈ a, v ∈ b}. Note that the neutral elements in A are 0 = ∅ and 1 = {(0, 0)}. For a ∈ A and n ∈ N we write a(n) = {(nu1 , nu2 ) | (u1 , u2 ) ∈ a}. The set A+ = {a ∈ A | (0, 0) ∈ / a} forms an ideal of the semi-ring A. For a, b ∈ A+ we define a ∗ b ∈ A+ by the formula (1 + a)(1 + b) = 1 + a ∗ b. For a ∈ A+ and n ∈ N we write a∗n = a ∗ · · · ∗ a for the n-fold power with respect to ∗. For a ∈ A and q ∈ N≥2 we define the Dirichlet polynomial X ξa,q (s) = q m−ns . (m,n)∈a Remark 2.7. (1) Let a, b ∈ A+ and n ∈ N. The following properties are immediate from the definitions: a(n) ⊂ a∗n , ξa,qn = ξa(n) ,q , and there exists C = C(a, b) ∈ R such that (1 + ξa,q )(1 + ξb,q ) − 1 ∼C ξa∗b,q for all q ∈ N≥2 . For instance, C(a, b) = 1 + min{|a|, |b|} works. Furthermore, if a ⊂ b, then ξa,q .1 ξb,q for all q ∈ N≥2 . (2) Let a, b ∈ A. LetSN(a) denote the “north-west”-Newton polytope associated to a, i.e., the convex hull of {u + (R≤0 × R≥0 ) | u ∈ a} in R2 . Then N(a) ⊂ N(b) if and only if there exists C ∈ R such that ξa,q (s) .C ξb,q (s) for all q ∈ N≥2 . In fact, if N(a) ⊂ N(b) one can take C = |a|. Q −1 Recall that the Dedekind zeta function ζK (s) = p∈Spec(O) (1 − |O/p|−s) of a number field K has abscissa of convergence 1, and that, for any subset T ⊂ Spec(O) with positive Q −1 analytic density, the abscissa of convergence of p∈T (1 − |O/p|−s) is equal to 1. Of course, every co-finite subset of Spec(O) has positive analytic density. Theorem 2.8. Let K be a number field with ring of integers OK , and let G be an affine group scheme over OK whose generic fiber is connected, simply connected semi-simple. Then there exist ◦ c(G) ∈ A+ , ◦ for every finite extension L of K with ring of integers OL , subsets R(L) ⊂ T (L) ⊂ Spec(OL ) with T (L) co-finite and R(L) of positive analytic density in Spec(OL ), ◦ for every ε ∈ R>0 , a constant C(ε) ∈ R>0 such that for every finite extension L of K with ring of integers OL the following hold. (1) For every q ∈ T (L), there is a subset c ⊂ c(G) such that, for every ε ∈ R>0 , ζG(OL,q ) − 1 ∼C(ε) ξc,|OL/q| for σ > α(G(OL,q)) + ε. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 11 (2) For every q ∈ R(L) and every ε ∈ R>0 , ζG(OL,q ) − 1 ∼C(ε) ξc(G),|OL/q| for σ > α(G(OL,q )) + ε. We remark that the element c(G) in Theorem 2.8 depends on G, but is not canonically determined by it: in the course of the proof we implicitly make a number of choices, influenced by triangulations of polytopes and resolutions of singularities, and different choices result in different elements c(G). In addition, the proof actually shows that the set R(L) is a Chebotarev set in the sense of Definition 3.14. The proof of Theorem 2.8 occupies a large part of the paper and is completed at the end of Section 6. We now show how Theorem 2.8 implies Theorem 1.5. Lemma 2.9. Let a ∈ A+ r {∅}, let L be a number field with ring of integers OL and R ⊂ Spec(OL ) of positive analytic density. Then the abscissa of convergence of the Q Dirichlet series ξ = q∈R (1 + ξa,|OL/q| ) is equal to max{(m + 1)/n | (m, n) ∈ a}. Proof. For any two sequences (xn )n∈N and (ynQ )n∈N of positive realQnumbers, the product Q n∈N (1 + xn + yn ) converges if and only if n∈N (1 + xn ) and n∈N (1 + yn ) converge individually. Thus, if a = b+ c in A, then the abscissa of convergence of ξ is the maximum Q Q of the abscissae of convergence of q∈R (1 + ξb,|OL/q| ) and q∈R (1 + ξc,|OL/q| ). Thus we may assume that a = {(m, n)}Qis a singleton. Let ζL,Spec(OL )\R (s) = q∈R (1 − |OL /q|−s )−1 denote the “restriction” of the Dedekind zeta function of L to the Euler product over factors in R. We observe that ξ(s) = ζL,Spec(OL )\R (ns − m) . ζL,Spec(OL )\R (2(ns − m)) Since the abscissa of convergence of ζL,Spec(OL )\R (s) is equal to 1 and ζL,Spec(OL )\R (s) 6= 0 for Re(s) > 1, we deduce that the abscissa of convergence of ξ is equal to (m + 1)/n.  Proof of Theorem 1.5. Let c(G) ∈ A+ , R(L) ⊂ T (L) ⊂ Spec(OL ) and C(ε), for ε > 0,
cL ) = be as in Theorem 2.8. As noted before,
Theorem 2.2 implies that α G(O
Q Q α q∈Spec(OL ) G(OL,q ) = α q∈T (L) G(OL,q ) . Moreover, R(L) ⊂ T (L) implies that Y  Y  (2.3) α G(OL,q ) ≤ α G(OL,q ) . q∈R(L) q∈T (L) For q ∈ R(L) and ε > 0, Theorem 2.8 yields that ζG(OL,q ) − 1 ∼C(ε) ξc(G),|OL/q| for σ > α(G(OL,q)) + ε. Lemma Q 2.5 implies that the left-hand side of (2.3) is equal to the abscissa of convergence of q∈R(L) (1 + ξc(G),|OL/q| ). Similarly, since ζG(OL,q ) − 1 .C(ε) ξc,|OL/q| .1 ξc(G),|OL/q| for σ > α(G(OL,q )) + ε, for every q ∈ T (L) and suitable c ⊂ c(G), the right-hand side of (2.3) is less than or Q equal to the abscissa of convergence of q∈T (L) (1 + ξc(G),|OL/q| ). Since R(L) has positive analytic density,
the two abscissae are equal. By Lemma 2.9 their common value, and cL ) , is completely determined by c(G). hence α G(O  12 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Proof of Theorem 1.1. Let S be a Chevalley group with root system Φ, i.e., an affine group scheme over Z whose generic fiber is split, connected, simply connected absolutely almost simple with absolute root system Φ. Consider an arithmetic group G(OS ) with the wCSP, as in the statement of the theorem. There is a finite extension L of K such that G and S are isomorphic over L. Denoting the ring of integers of L by OL , Theorems 1.4 and 1.5 imply that



b = α G(O cL ) = α S(O cL ) = α S(Z) b . α(G(OS )) = α G(O) Thus α(G(OS )) depends only on Φ.  Proof of TheoremQ1.2. Let H be a semi-simple group in characteristic 0. Recall that this means that H = rj=1 Hj (Fj ), where each Fj is a local field of characteristic 0, and each Hj is a connected, almost simple group defined over Fj . Let Γ be an arithmetic irreducible lattice in H. Then there are a number field K with ring of integers O, a finite set S of places of K, an affine group scheme G defined over OS whose generic fiber is connected, simply connected Q absolutely almost simple, and a continuous homomorphism ψ : v∈S G(Kv ) → H whose kernel and cokernel are compact such that ψ(G(OS )) is commensurable to Γ. Since ker(ψ) ∩ G(OS ) is finite and G(OS ) is residually finite, the latter contains a finite-index subgroup that is isomorphic to a finite-index subgroup of Γ. Hence α(Γ) = α(G(OS )). Fix j ∈ {1, . . . , r}, and denote by Lie(G) and Lie(HL j ) the Lie algebras associated to G and Hj . The homomorphism ψ induces a surjection v∈S Lie(G)(Kv ) → Lie(Hj )(Fj ) which is additive and preserves Lie brackets; if Fj is non-archimedean of residue characteristic p, the construction uses the Lie correspondence for p-adic groups. Taking the tensor product with C – over R if Fj is archimedean and over Qp embedded into C if Fj is non-archimedean of residue characteristic p – we obtain a Lie algebra epimorphism Lie(G)(C)m → Lie(Hj )(C)n , for some m, n ∈ N. In particular, since G(C) is almost simple, the group G(C) and the simple factors of the groups Hj (C) are all isogenous to one another, and therefore have the same absolute root system Φ. The claim now follows from Theorem 1.1.  As mentioned above, the abscissa of convergence for groups is a commensurability invariant; see [30, Lemma 2.2]. We conclude the section with a more general lemma about relative zeta functions. Definition 2.10. Let G be a group with a normal subgroup N ⊂ G and let ϑ be an irreducible, finite-dimensional complex representation of N. Denote by Irr(G|ϑ) the set of (equivalence classes of) finite-dimensional irreducible complex representations ̺ of G such that ϑ is a constituent of the restriction ResG N ̺, which is completely reducible by Clifford’s Theorem. For n ∈ N we denote by Rn (G|ϑ) the number of representations ̺ ∈ Irr(G|ϑ) such that dim ̺ ≤ n dim ϑ. Suppose that Rn (G|ϑ) is finite for all n ∈ N. Then the relative zeta ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 13 function of G over ϑ is defined as the Dirichlet generating series X  dim ̺ −s ζG|ϑ (s) = . dim ϑ ̺∈Irr(G|ϑ) We also set ′ ζG|ϑ (s) = X ̺∈Irr(G|ϑ) dim ̺>dim ϑ  dim ̺ dim ϑ −s . Lemma 2.11. Let N ⊂ H ⊂ G be groups such that |G : H| is finite and N is normal in G. Let ϑ ∈ Irr(N), and suppose that Rn (G|ϑ), Rn (H|ϑ) are finite for all n ∈ N. (1) Let m ∈ N. Then |G : H|−1R⌊m/|G:H|⌋ (H|ϑ) ≤ Rm (G|ϑ) ≤ |G : H|Rm (H|ϑ). (2) Let σ ∈ R≥0 . If one of ζG|ϑ (σ) or ζH|ϑ (σ) converges, then so does the other, and |G : H|−1−σ ζH|ϑ (σ) ≤ ζG|ϑ (σ) ≤ |G : H|ζH|ϑ (σ). (3) Suppose that min{dim ̺/ dim ϑ | ̺ ∈ Irr(G|ϑ) with dim ̺ > dim ϑ} > |G : H|. Let ′ ′ σ ∈ R≥0 . If one of ζG|ϑ (σ) or ζH|ϑ (σ) converges, then so does the other, and ′ ′ ′ |G : H|−1−σ ζH|ϑ (σ) ≤ ζG|ϑ (σ) ≤ |G : H|ζH|ϑ (σ). Proof. Consider the bipartite graph B whose vertex set is Irr(G|ϑ) ⊔ Irr(H|ϑ) and which has the property that there is an edge between ̺1 ∈ Irr(G|ϑ) and ̺2 ∈ Irr(H|ϑ) if and only if ̺2 is a constituent of ResG H ̺1 . By Nakayama’s generalisation of Frobenius reciprocity, the latter condition is equivalent to ̺1 being a constituent of IndG H ̺2 ; see [21, Chapter VII, Section 4]. This implies that ◦ the degree of every vertex of B is positive and bounded by |G : H| and that ◦ if ̺1 ∈ Irr(G|ϑ) and ̺2 ∈ Irr(H|ϑ) are connected by an edge, then dim ̺2 ≤ dim ̺1 ≤ |G : H| dim ̺2 . Let m ∈ N. Write Irr(G|ϑ)m ⊂ Irr(G|ϑ) for the subset consisting of representations of dimension at most m dim ϑ, and likewise define Irr(H|ϑ)m . Then Irr(G|ϑ)m is contained in the set of neighbors of Irr(H|ϑ)m , hence Rm (G|ϑ) = |Irr(G|ϑ)m | ≤ |G : H| |Irr(H|ϑ)m | = |G : H|Rm (H|ϑ). Similarly, Irr(H|ϑ)⌊m/|G:H|⌋ is contained in the set of neighbors of Irr(G|ϑ)m , hence R⌊m/|G:H|⌋ (H|ϑ) = |Irr(H|ϑ)⌊m/|G:H|⌋ | ≤ |G : H| |Irr(G|ϑ)m | = |G : H|Rm (G|ϑ). This proves (1). A similar argument yields (2) and (3). For (3) one observes that, if min{dim ̺/ dim ϑ | ̺ ∈ Irr(G|ϑ) with dim ̺ > dim ϑ} > |G : H| and if ̺1 ∈ Irr(G|ϑ) and ̺2 ∈ Irr(H|ϑ) are connected by an edge, then dim ̺1 > dim ϑ if and only if dim ̺2 > dim ϑ.  14 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL 3. Base Change for Finite Groups of Lie Type In this section we prove Theorem 3.1, a variant of Theorem 2.8 for zeta functions of semi-simple algebraic groups over finite fields. In Section 3.2 we apply our results to finite quotients of arithmetic groups. 3.1. Finite Groups of Lie Type. Given a root system Φ, we denote by rk Φ its rank and by Φ+ a choice of positive roots. By a Lie type L we mean a pair (Φ, τ ), where Φ is a root system with an automorphism τ stabilising Φ+ . We say that a reductive algebraic group G defined over a finite field Fq has Lie type L = (Φ, τ ) if the absolute root system of G associated to an Fq -rational and maximally Fq -split maximal torus of G is Φ and the action of the Frobenius automorphism Frobq on Φ is given by τ ; compare [13, Chapter 3]. For each finite field Fq , the connected semi-simple algebraic groups over Fq are parametrized up to isogeny by their Lie types. Given a Lie type L = (Φ, τ ) we write Lsp = (Φ, Id) for the Lie type of a split group with underlying root system Φ. Recall the notation A+ and ξa,q from Definition 2.6. In this section we prove the following result. Theorem 3.1. Let Φ be a non-trivial root system, and let LΦ denote the collection of Lie types with underlying root system Φ. Let Q denote the set of all prime powers. Then there exist C ∈ R, m ∈ N, a(Φ) ∈ A+ , and a(L, q) ∈ A+ for (L, q) ∈ LΦ × Q such that the following hold: (1) a(L, q) ⊂ a(Φ) for all (L, q) ∈ LΦ × Q, (2) a((Φ, Id), q) = a(Φ) for all q ≡m 1, (3) for every connected semi-simple algebraic group G defined over a finite field Fq which has Lie type L ∈ LΦ , ζG(Fq ) − |G(Fq )/[G(Fq ), G(Fq )]| ∼C ξa(L,q),q . Moreover, (rk Φ, |Φ+ |) ∈ a(Φ), and every connected semi-simple algebraic group G defined over a finite field Fq with absolute root system Φ satisfies (3.1) ζG(Fq ) (s) ∼C 1 + q rk Φ−|Φ + |s . Remark 3.2. We remark that if G is a connected, simply connected semi-simple algebraic group over Fq and q > 3, then G(Fq ) is perfect and therefore |G(Fq )/[G(Fq ), G(Fq )]| = 1. Indeed, for simple groups this is a result of Tits [42] (see also [34, Theorem 24.17]), and the semi-simple case follows by taking products. Example 3.3. The representations of the special linear group SL2 (Fq ) over a finite field Fq are well known; e.g. see [13, Chapter 15]. In particular, we have ζSL2 (Fq ) (s) − 1 ∼2 q 1−s for sufficiently large q. Consider the following semi-simple algebraic groups defined over Fq whose absolute root systems are both A1 × A1 : G1 = SL2 × SL2 and G2 = RFq2 |Fq SL2 , the restriction of scalars of SL2 defined over a quadratic extension. Then G1 (Fq ) = SL2 (Fq ) × SL2 (Fq ) and ζG1 (Fq ) (s) − 1 ∼4 q 1−s + q 2−2s , whereas G2 (Fq ) = SL2 (Fq2 ) and ζG2 (Fq ) (s) − 1 ∼2 q 2−2s . Since q 1−s + q 2−2s 6∼C q 2−2s for any fixed C but unbounded q, we see that the inclusion in part (1) of Theorem 3.1 can be strict. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 15 Whilst Theorem 3.1 is a result on semi-simple groups, it leads to the following corollary on reductive groups. Corollary 3.4. Let G be a connected reductive algebraic group defined over a finite field Fq with absolute root system Φ. There is a constant D ∈ R, depending only on Φ and the + dimension of G, such that ζG(Fq ) (s) ∼D q dim Z(G) (1 + q rk Φ−|Φ |s ). The proof of Theorem 3.1 and its Corollary 3.4 is given in Section 3.1.4, and prepared in the preceding sections. It is based on Lusztig’s classification of irreducible representations of finite groups of Lie type; e.g., see [13, Chapter 13]. We write G∗ for the dual group of a connected reductive group G and recall that, while G and G∗ have isomorphic Weyl groups W ∼ = W ∗ , the absolute root system Φ∨ of G∗ is dual to Φ in the sense that the roles of long and short roots are interchanged. We also note that G is simply connected respectively adjoint if and only if G∗ is adjoint respectively simply connected. Since equivalence classes of irreducible representations of G(Fq ) are parametrized by the corresponding characters, we use the notation Irr(G(Fq )) in a flexible way to denote also the set of irreducible complex characters of G(Fq ). The set Irr(G(Fq )) is the disjoint union of certain Lusztig series E(G(Fq ), (g)) so that X X (3.2) ζG(Fq ) (s) = χ(1)−s , (g)⊂G∗ (Fq ) χ∈E(G(Fq ),(g)) where the outer sum ranges over G∗ -conjugacy classes of semi-simple elements in G∗ (Fq ). 3.1.1. Unipotent Zeta Functions. The elements of E(G(Fq ), (1)) are known as the unipotent characters of G(Fq ). We set X unip ζG(F (s) = χ(1)−s . q) χ∈E(G(Fq ),(1)) Proposition 3.5. Let Φ be a non-trivial root system and let LΦ denote the collection of Lie types with underlying root system Φ. Then there are C ∈ R and bc (L) ∈ A+ for L ∈ LΦ such that the following hold: (1) bc (L) ⊂ bc (Lsp ) for all L ∈ LΦ , (2) for every connected reductive algebraic group G defined over a finite field Fq which has Lie type L ∈ LΦ , unip ζG(F − 1 ∼C ξbc (L),q q) and unip ζG(F ∼C 1. q) Proof. Throughout, all constants – be they real or elements of A+ – depend only on Φ or a given Lie type L ∈ LΦ . Let G be a connected reductive algebraic group of Lie type L = (Φ, τ ) ∈ LΦ . Then G/Z(G) is a semi-simple group over Fq of adjoint type and, by [13, Proposition 13.20], every unipotent character of G(Fq ) factors through (G/Z(G))(Fq ). Therefore we may assume that G is semi-simple of adjoint type and thus a direct product of Fq -simple groups Si of Lie type Li = (Φi , τi ), say, where i ∈ {1, . . . , m}. The unipotent characters of G(Fq ) = S1 (Fq ) × · · · × Sm (Fq ) are the irreducible characters 16 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL G that appear in the Deligne–Lusztig character RT (1), where T is a maximal torus defined G over Fq . The generalised character RT (1) arises from the action of G(Fq ) × T(Fq ) on the cohomologies of the Deligne–Lusztig variety X = ℓ−1 (U), where ℓ denotes the Lang map and U is the unipotent radical of a Borel subgroup containing T; see [13, Section 11]. This construction is compatible with taking products, hence X can be taken to be the product of the Deligne–Lusztig varieties of the groups Si (Fq ). The Künneth formula now implies that unip ζG(F q) (3.3) = m Y ζSunip . i (Fq ) i=1 Fix i ∈ {1, . . . , m}. Since Si is Fq -simple, τi permutes the irreducible components Φi,1 , n(i) Φi,2 , . . . , Φi,n(i) of Φi transitively. Thus τi restricts to an automorphism of Ψi = Φi,1 . ei = e i for the adjoint absolutely simple algebraic group over Fqn(i) of Lie type L Writing S n(i) ei (Fqn(i) ). By [6, Sections 13.8 and 13.9], the number k(i) of (Ψi , τi ), we have Si (Fq ) = S e i . This shows that ζ unip .C 1 for a e i (Fqn(i) ) depends only on L unipotent characters of S i Si (Fq ) suitable constant Ci ∈ R. Since the trivial character is unipotent, we also get 1 .1 ζSunip . i (Fq ) unip This yields ζSunip ∼Ci 1 and thus ζG(F ∼C1 ···Cm 1, using (3.3). q) i (Fq ) Furthermore, looking through the tables in [6, Sections 13.8 and 13.9], one sees that e i such that the degrees there are polynomials fi,1 , . . . , fi,k(i) ∈ Q[X] depending only on L ei (Fqn(i) ) are precisely fi,1 (q n(i) ), . . . , fi,k(i) (q n(i) ). Moreover, of the unipotent characters of S only one of these polynomials has degree zero, namely the constant polynomial 1 giving the degree of the trivial character. We may assume that fi,1 = 1 and define e i ) = {(0, deg fi,j ) | 2 ≤ j ≤ k(i)} ∈ A+ . bc (L e i and not only We show below that, while the degrees of the fi,j generally depend on L sp e = (Ψi , Id) is a superset of the set of degrees on Ψi , the set of degrees in the split case L i e i )(n(i)) in the twisted cases. Granted that this is so, we continue and define bc (Li ) = bc (L so that for a suitable constant Ci′ ∈ R we obtain (3.4) ζSunip − 1 = ζSeunip i (Fq ) (F i q n(i) ) − 1 ∼Ci′ ξbc (Le i ),qn(i) = ξbc (Li ),q . We set bc (L) = bc (L1 ) ∗ · · · ∗ bc (Lm ). From our claim regarding the degrees of the polynomials fi,j and Remark 2.7 we deduce that, for each i ∈ {1, . . . , m}, (3.5) e i )(n(i)) ⊂ bc (Ψi , Id)(n(i)) ⊂ bc (Ψi , Id)∗n(i) = bc (Lsp ). bc (Li ) = bc (L i In the special case m = 1 this establishes assertion (1) of the proposition, and (3.4) directly yields the remaining first assertion in (2). In the general case we conclude the proof based on (3.3) and Remark 2.7. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 17 It remains to justify the claim about the degrees of unipotent characters for split and twisted finite groups of Lie type with the same underlying absolute root system. Simpliei ) be a connected, adjoint absolutely fying the notation used above, let S (rather than S simple algebraic group defined over Fq (rather than Fqn(i) ), and let Ssp denote a split form of S over Fq . We consider the possible groups case by case. By the remarks following [13, Theorem 3.17], the relevant twisted Lie types are 2 An , 2 Dn , 3 D4 , and 2 E6 , where the left index represents as customary the order of the automorphism of the root system. (In our setup, the Suzuki and Ree groups do not occur. In fact, the analogous statement for groups of Lie type F4 and 2 F4 is incorrect; see Remark 3.6.) Case 1: S has Lie type 2 An and Ssp has Lie type An . The unipotent characters of each group, S(Fq ) and Ssp (Fq ), are parametrized by partitions α of n + 1. The degree χα (1) of the unipotent character χα associated to α in either case is given by a polynomial in q whose degree only depends on α; see [6, p. 465]. This proves that bc (2 An ) = bc (An ). Case 2: S has Lie type 2 Dn and Ssp has Lie type Dn . The unipotent characters of Ssp (Fq ) are parametrized by so-called Lustzig symbols (or just “symbols” in the terminology of [6]). These are certain pairs (S, T ) of finite subsets of Z≥0 such that |S| − |T | ≡ 0 (mod 4), where pairs of the form (S, S) correspond to pairs of characters rather than single characters, but this subtlety is irrelevant for our purposes. Similarly, the unipotent characters of S(Fq ) are parametrized by pairs (S, T ) such that |S| − |T | ≡ 2 (mod 4). Let (S, T ) be a pair of the second kind and suppose without loss of generality that there exists an element a ∈ S r T . Then (S ′ , T ′) = (S r {a} , T ∪ {a}) is a pair of the first kind. Inspection of the formulae giving the degrees χ(S,T ) (1) and χ(S ′ ,T ′ ) (1) of the unipotent characters χ(S,T ) of S(Fq ) associated with (S, T ) and χ(S ′ ,T ′ ) of Ssp (Fq ) associated with (S ′ , T ′ ) yields: the polynomials in q giving χ(S,T ) (1) and χ(S ′ ,T ′ ) (1) have equal degrees. For details see [6, p. 471 and 475f.]. This proves that bc (2 Dn ) ⊂ bc (Dn ). Case 3: S has Lie type 3 D4 and Ssp has Lie type D4 . The set of degrees of the polynomials expressing the degrees of unipotent characters of S(Fq ) is {0, 5, 9, 11, 12} (see [6, p. 478]), whereas the corresponding set for Ssp (Fq ) is {0, 5, 6, 9, 10, 11, 12}. This shows that bc (3 D4 ) ⊂ bc (D4 ). Case 4: S has Lie type 2 E6 and Ssp has Lie type E6 . The 30 unipotent characters of these groups are in degree-preserving bijective correspondence; cf. [6, p. 480f.]. This shows that bc (E6 ) = bc (2 E6 ).  Remark 3.6. Proposition 3.5 extends only partly to the remaining groups of Lie type, the Suzuki and Ree groups. For the groups B2 (q) and 2 B2 (q 2 ) the degrees of the polynomials giving the degrees of the unipotent characters coincide. The same is true for the groups G2 (q) and 2 G2 (q 2 ). But the degrees of the polynomials giving the unipotent character degrees of the Ree groups 2 F4 (q 2 ) are {0, 9, 14, 20, 21, 22, 24} (see [6, p. 489]), whereas the corresponding degrees for the split groups F4 (q) are {0, 11, 15, 20, 21, 22, 23, 24} (see [6, p. 479]). 18 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Remark 3.7. In fact, the sets bc (L) in Proposition 3.5 are all contained in {0} × Z>0 , so ξbc (L),q could be replaced by q −m(L)s , where m(L) = min{m ∈ Z>0 | (0, m) ∈ bc (L)}. Moreover, it is conceivable that m(L) actually only depends on the root system Φ. As the proof of the proposition shows, this reduces to a question regarding the degrees of unipotent characters of groups of type Dn . More precisely, it would follow if the degrees of the polynomials giving the degrees of the unipotent characters of the (untwisted) groups of type Dn indexed by special Lusztig symbols of the form (S, S) were already among the corresponding degrees obtained from Lusztig symbols of the form (S, T ) with S 6= T . 3.1.2. Connected Centralizers of Semi-Simple Elements. In preparation for the proof of Theorem 3.1 we record some auxiliary facts about the connected centralisers of semisimple elements in algebraic groups over finite fields. Let Φ be a root system. We consider a connected split reductive algebraic group G over a finite field Fq whose absolute root system, associated to some Fq -rational split maximal torus T, is isomorphic to and identified with Φ. We denote by Falg q the algebraic ◦ closure of Fq . The connected centraliser CG (g) of any semi-simple element g ∈ G(Falg q ) is a reductive subgroup of maximal rank in G. Indeed, every semi-simple element of G is conjugate to an element of T. Furthermore the connected centraliser of g ∈ T(Falg q ) is the reductive group hT, Uα | α ∈ Ψg i with root system Ψg = {α ∈ Φ | α(g) = 1}, where each Uα denotes the root subgroup associated to α; see [13, Proposition 2.3]. For every extension E of Fq we put (3.6) C(G, T, E) = {Ψg | g ∈ T(E)}. Furthermore, we set (3.7) C(Φ) = [ C(G, T, Fq ), G,T,Fq where G, T and Fq range over all possible choices for the fixed root system Φ. In Proposition 3.9 we show that, in fact, C(G, T, Fq ) = C(Φ) as long as Fq contains a sufficient supply of roots of unity. Lemma 3.8. Let ϕ : G → H be an isogeny, i.e., a surjective morphism with finite central kernel, between algebraic groups defined over a finite field Fq . If g ∈ G(Fq ), then ϕ maps CG (g)◦ onto CH (ϕ(g))◦ . Proof. Write h = ϕ(g) and observe that \ (3.8) CG (g̃) = CG (g) ⊂ ϕ−1 (CH (h)). g̃∈G(Falg q ), ϕ(g̃)=h T Furthermore, ϕ−1 (CH (h))/ ϕ(g̃)=h CG (g̃) acts faithfully by conjugation on the finite set g ker(ϕ). This shows that the inclusion in (3.8) is of finite index, and hence CG (g)◦ is a finite index subgroup of ϕ−1 (CH (h)). Since ϕ(CG (g)◦) is connected and of finite index in CH (h), we conclude that ϕ(CG (g)◦) = CH (h)◦ .  ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 19 Proposition 3.9. Let Φ be a root system. There exists m ∈ N such that for every connected split reductive algebraic group G over a finite field Fq whose absolute root system, associated to an Fq -rational split maximal torus T, is isomorphic to and identified with Φ the following holds: if Fq contains primitive mth roots of unity, equivalently q ≡m 1, then C(G, T, Fq ) = C(Φ). In particular, if G is as above and q ≡m 1, then for every semi-simple element g ∈ ◦ ◦ G(Falg q ), there is a semi-simple element g0 ∈ G(Fq ) such that CG (g) and CG (g0 ) are G-conjugate. Proof. Fix Ψ ∈ C(Φ). Clearly, it suffices to show that there exists m(Ψ) ∈ N so that Ψ arises as the absolute root system of a connected centraliser in each instance of (G, T, Fq ) with q ≡m(Ψ) 1. Let G be a connected split reductive algebraic group over a finite field Fq , with absolute root system Φ. The group G is the image of an isogeny from the direct product of the connected split semi-simple group [G, G] and Z(G)◦ , a torus. Using Lemma 3.8, we may concentrate on the case that G is semi-simple and hence an isogenous image of a direct product of split simply-connected almost simple algebraic groups. Applying once more Lemma 3.8, we may restrict to the case that G is simply-connected almost simple, with root datum (X, Φ, ZΦ∨ , Φ∨ ), say. The root lattice ZΦ is a sublattice of finite index in the free Z-lattice X. We fix free generators χ1 , . . . , χℓ of X. Let T ⊂ G be an Fq -rational split maximal torus. We identify Φ with the root system and X with the character group associated to T. Let g ∈ T(Falg q ). As noted before, the connected centraliser of g is the reductive subgroup CG (g)◦ = hT, Uα | α ∈ Ψg i ∗ with root system Ψg = {α ∈ Φ | α(g) = 1}. Furthermore, the multiplicative group (Falg q ) is isomorphic to the additive group Qp′ /Z, where p denotes the characteristic of Fq and Qp′ the ring of all rational numbers whose denominator is not divisible by p. Under this isomorphism F∗q corresponds to the additive group Z[1/(q − 1)]/Z. The conditions α(g) = 1 for α ∈ Ψg and α(g) 6= 1 for α ∈ Φ r Ψg , which determine CG (g)◦, translate into a finite system of linear equations and inequations with integer coefficients in ℓ variables x1 , . . . , xℓ , corresponding to the generators χ1 , . . . , χℓ of X. The element g corresponds to a solution (a1 /b1 + Z, . . . , aℓ /bℓ + Z) ∈ (Qp′ /Z)ℓ of the system modulo Z. Without loss of generality, we may assume that gcd(ai , bi ) = 1 for each i ∈ {1, . . . , ℓ}, and we set m(Ψg ) = lcm(b1 , . . . , bℓ ). Then g ∈ T(Fq ) if and only if Fq contains primitive roots of unity of degree m(Ψg ), equivalently q ≡m(Ψg ) 1. As Ψ ∈ C(Φ), there is an instance of (G, T, Fq ) and g ∈ Fq as described above such that Ψ = Ψg . This single solution shows that Ψ ∈ C(G, T, Fq ) for all instances (G, T, Fq ) with q ≡m(Ψ) 1.  Corollary 3.10. There exists c ∈ N, depending only on Φ, such that, for all finite fields Fq of characteristic p > c, C(G, T, Falg q ) = C(Φ). 20 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Remark 3.11. Proposition 3.9 is related to more detailed investigations into semi-simple conjugacy classes and their centralizers in finite groups of Lie type, which were initiated by Carter and Deriziotis. In particular, their results imply that C(G, T, Falg q ) = C(Φ) if the characteristic of Fq is greater than 5; see [12, Proposition 2.3 and remarks]. For an overview, including also subsequent developments we refer to [20, Chapters 2 and 8] and the references given therein. Corollary 3.12. Let Φ be a non-trivial root system, and let C(Φ) be as defined in (3.7). Let W (Φ) denote the Weyl group of Φ. Let G be a connected semi-simple algebraic group defined over a finite field Fq with absolute root system Φ. Identify Φ with the root system associated to a maximal torus T of G, and let C(G, T, Falg q ) be as defined in (3.6). Then there are connected subgroups HΨ of G, not necessarily defined over Fq and indexed by Ψ ∈ C(G, T, Falg q ), such that the following hold. (1) The groups HΨ form a – typically redundant – set of representatives for the Gconjugacy classes of connected centralisers CG (g)◦ of semi-simple elements g ∈ G(Falg q ). (2) The data (dim Z(HΨ ), dim HΨ , |Ψ+ |) depend only on Φ and Ψ; more precisely, the first two entries satisfy dim Z(HΨ ) = rk Φ − rk Ψ and dim HΨ = rk Ψ + 2|Ψ+ |. (3) For every Ψ, the number of G(Fq )-conjugacy classes of connected centralisers CG (g)◦ ⊂ G of semi-simple elements g ∈ G(Fq ) that are G-conjugate to HΨ is at most |W (Φ)||C(Φ)|. ◦ Proof. For each Ψ ∈ C(G, T, Falg q ), choose HΨ as the connected centraliser CG (gΨ ) of a semi-simple element gΨ ∈ T(Falg q ) such that the root system associated to HΨ is Ψ. Clearly, (1) and (2) are satisfied. It remains to justify (3). Fix Ψ ∈ C(G, T, Falg q ). We may assume that H = HΨ is itself equal to CG (g0 )◦ for a semi-simple element g0 ∈ G(Fq ). Let T0 be an Fq -defined maximal torus of H. We observe that g0 ∈ Z(H◦ ) ⊂ T0 (see [6, Proposition 3.5.1]) and that T0 is also an Fq -defined maximal torus of G. By [6, Proposition 3.3.3], the number of G(Fq )-conjugacy classes of Fq -defined maximal tori is at most |W (Φ)|. Thus we only need to bound the number of connected centralisers of semi-simple elements g in any fixed Fq -defined maximal torus, but this number is clearly bounded by |C(Φ)|.  3.1.3. Non-Central Semi-Simple Conjugacy Classes. Let G be a connected reductive algebraic group defined over a finite field Fq and let G∗ be the dual group. By [6, Proposition 3.6.8], we have Z(G∗ (Fq )) = Z(G∗ )(Fq ) so that central geometric conjugacy classes and central rational conjugacy classes in G∗ (Fq ) coincide. We set X X nc (3.9) ζG(F (s) = χ(1)−s , q) (g)6⊂Z(G∗ (Fq )) χ∈E(G(Fq ),(g)) ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 21 where the outer sum ranges over the non-central semi-simple G∗ -conjugacy classes (g) in G∗ (Fq ). For a non-trivial root system Φ, we define  (3.10) bnc (Φ) = (rk(Φ) − rk(Ψ), |Φ+ | − |Ψ+ |) | Ψ ∈ C(Φ) r {Φ} ∈ A+ . Lemma 3.13. Let Φ be a non-trivial root system and let GΦ denote the collection of pairs (G, q), where G is a connected semi-simple algebraic group over a finite field Fq with absolute root system Φ. Then there are C ∈ R and bnc (G, q) ⊂ bnc (Φ) for (G, q) ∈ GΦ such that for all (G, q), (H, q) ∈ GΦ the following hold: (1) bnc (G, q) ⊂ bnc (H, q) whenever there is an isogeny ϕ : G → H defined over Fq , (2) bnc (G, q) = bnc (Φ) whenever G is split and q ≡m 1, where m is as in Proposition 3.9, nc (3) ζG(F ∼C ξbnc (G,q),q , q) (4) (rk Φ, |Φ+ |) ∈ bnc (Φ), and 1 + ξbnc (G,q),q (s) ∼C 1 + q rk Φ−|Φ + |s . Proof. Consider (G, q) ∈ GΦ and let G∗ be the dual group of G. The centralizer CG∗ (g) of a semi-simple element g ∈ G∗ (Fq ) is a reductive subgroup of maximal rank in G∗ . Furthermore, there is a surjection ψg : E(G(Fq ), (g)) → E(CG∗ (g)(Fq ), (1)) such that the fibers of ψg have sizes at most |Z(G)|, and the degree of an element χ ∈ E(G(Fq ), (g)) is given by χ(1) = |G∗ (Fq ) : CG∗ (g)(Fq )|q′ · (ψg (χ))(1), where nq′ denotes the prime-to-q part of a number n; see [13, Theorem 13.23 and Remark 13.24] and [6, Proposition 4.4.4] in the case where G has trivial centre, and [32, Proposition 5.1] for the general case. It follows that X unip nc ζG(F (s) ∼ |G∗ (Fq ) : CG∗ (g)(Fq )|−s |Z(G)| q ′ · ζCG∗ (g)(Fq ) (s). q) (g)6⊂Z(G∗ (Fq )) Consider a semi-simple element g ∈ G∗ (Fq ). The index |CG∗ (g) : CG∗ (g)◦ | is bounded by a constant C1 ∈ R, depending only on Φ; see [13, Remark 2.4]. This gives −s ◦ ∗ |G∗ (Fq ) : CG∗ (g)(Fq )|−s q ′ ∼C1 |G (Fq ) : CG∗ (g) (Fq )|q ′ . Moreover, the unipotent characters of CG∗ (g)(Fq ) are the irreducible characters whose restrictions to CG∗ (g)◦(Fq ) are sums of unipotent characters. From Proposition 3.5 we conclude that there is a constant C2 ∈ R such that ζCunip (s) ∼C2 1. G∗ (g)(Fq ) We get that nc ζG(F q) ∼|Z(G)|C1 C2 X (g)6⊂Z(G∗ (Fq ))  |G∗ (Fq )|q′ |CG∗ (g)◦ (Fq )|q′ −s . Applying Corollary 3.12 to G∗ , we obtain N ∈ N, bounded by |C(Φ)|, and algebraic subgroups H1 , . . . , HN ⊂ G∗ with the properties described in the corollary. For each i ∈ {1, . . . , N} denote the absolute root system of Hi by Ψi ∈ C(Φ). Given a non-central semisimple element g ∈ G∗ (Fq ), the group CG∗ (g)◦ is G∗ -conjugate to some Hi . There is a 22 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL + + constant C3 ∈ R such that |G∗ (Fq )|q′ ∼C3 q dim G−|Φ | and |CG∗ (g)◦ (Fq )|q′ ∼C3 q dim Hi −|Ψi | . Therefore + |G∗ (Fq )|q′ + dim G−|Φ+ |−dim Hi +|Ψ+ i | = q |Φ |−|Ψi | 2 q ∼ C 3 |CG∗ (g)◦(Fq )|q′ for all i ∈ {1, . . . , N}. Writing C4 = |Z(G)|C1C2 C32 N, we obtain nc (3.11) ζG(F (s) ∼C4 q) N X |{(g) 6⊂ Z(G∗ (Fq )) | CG∗ (g)◦ is G∗ -conjugate to Hi }| · q −(|Φ + |−|Ψ+ |)s i . i=1 The additional factor N of C4 is included, because CG∗ (g)◦ could be conjugate to more than one Hi . Denote by I(G, q) the set of all i ∈ {1, . . . , N} such that there exists a non-central semisimple element g ∈ G∗ (Fq ) with CG∗ (g)◦ being G∗ -conjugate to Hi . Fix i ∈ I(G, q) and let Ki,1 , . . . , Ki,n(i) be a selection of such centralisers, forming a complete set of representatives up to G∗ (Fq )-conjugacy. By Corollary 3.12, the number n(i) is uniformly bounded in terms of Φ. The collection of non-central semi-simple conjugacy classes (g) ⊂ G∗ (Fq ) such that CG∗ (g)◦ is G∗ -conjugate to Hi decomposes as a disjoint union as follows: (3.12) {(g) 6⊂ Z(G∗ (Fq )) | CG∗ (g)◦ is G∗ -conjugate to Hi } = n(i) G {(g) 6⊂ Z(G∗ (Fq )) | CG∗ (g)◦ is G∗ (Fq )-conjugate to Ki,j } . j=1 Fix also j ∈ {1, . . . , n(i)}. Now [29, Lemma 2.2(ii)] supplies a constant C5 ∈ R such that, for every non-central semi-simple conjugacy class (g) ⊂ G∗ (Fq ) with CG∗ (g)◦ being G∗ (Fq )-conjugate to Ki,j , 1 ≤ |{g x ∈ (g) | CG∗ (g x )◦ = Ki,j }| ≤ C5 . It follows that (3.13) |{(g) 6⊂ Z(G∗ (Fq )) | CG∗ (g)◦ is G∗ (Fq )-conjugate to Ki,j }| ∼C5 |{g ∈ G∗ (Fq ) | CG∗ (g)◦ = Ki,j }| . If g ∈ G∗ (Fq ) is semi-simple and CG∗ (g)◦ = Ki,j , then g ∈ Ki,j (Fq ). Choose a maximal torus Ti,j of Ki,j . The torus Ti,j is also maximal in G∗ . Denote the set of roots of (G∗ , Ti,j ) by Λi,j ⊂ Hom(Ti,j , Gm ) and the set of roots of (Ki,j , Ti,j ) by ∆i,j ⊂ Hom(Ti,j , Gm ). Note that Λi,j is isomorphic to Φ and ∆i,j is isomorphic to Ψi . The set of elements g ∈ Z(Ki,j )◦ with CG∗ (g)◦ = Ki,j is the complement of the union of the zero loci of the roots in Λi,j r ∆i,j . Each such zero locus is the extension of a proper sub-torus by a finite group. The order of that finite group, i.e., the number of connected components of ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 23 the zero locus, is bounded by some constant depending only on Φ; see, for example, [15, Corollary 9.7.9]. For every torus T, we have (3.14) 2− dim T q dim T ≤ |T(Fq )| ≤ 2dim T q dim T ; see [35, Lemma 3.5]. Hence there is a constant D ∈ R such that, for all sufficiently large q, (3.15) q dim Z(Ki,j ) .D | {g ∈ G∗ (Fq ) | CG∗ (g)◦ = Ki,j } |. By [15, Corollary 9.7.9], the index |Z(Ki,j ) : Z(Ki,j )◦ | for the connected reductive group Ki,j is bounded by some constant depending on Φ and, of course, dim Z(Kij ) = dim Z(Hi ). Using (3.12), (3.13), (3.14), and (3.15), it follows that there is a constant C6 ∈ R such that |{(g) 6⊂ Z(G∗ (Fq )) | CG∗ (g)◦ is G∗ -conjugate to Hi }| ∼C6 q dim Z(Hi ) , and from (3.11) we obtain nc ζG(F (s) ∼C4 C6 q) X q dim Z(Hi )−(|Φ + |−|Ψ+ |)s i . i∈I(G,q) Recalling from (3.10) the definition of bnc (Φ), we put  nc bnc (G, q) = (dim Z(Hi ), |Φ+ | − |Ψ+ i |) | i ∈ I(G, q) ⊂ b (Φ) so that nc ζG(F ∼C4 C6 N ξbnc(G,q),q . q) The extra factor N in the index of the ∼-symbol accommodates for the fact that Hi and Hj may lead to the same data even though i 6= j. This completes the proof of assertion (3) of the lemma. Next we prove (4). For i ∈ I(G, q), [29, Lemma 2.5] yields that (3.16) dim Z(Hi ) rk Φ rk G = + . + ≤ + + |Φ | |Φ | |Φ | − |Ψi | Moreover, we have dim Z(Hi ) ≤ rk G for each i ∈ I(G, q), and for sufficiently large q (depending only on Φ; see [6, Proposition 3.6.6]) there is at least one Hi , i ∈ I(G, q), namely a non-degenerate maximal torus, for which (3.17) + (dim Z(Hi ), |Φ+ | − |Ψ+ i |) = (rk Φ, |Φ |). From this we deduce that, for sufficiently large q, 1 + ξbnc (G,q),q (s) ∼N +1 1 + ξ{(rk Φ,|Φ+|)},q (s) = 1 + q rk Φ−|Φ + |s ; indeed, the inequality &N +1 is clear from (3.17), and .N +1 follows from Remark 2.7, noting that |bnc (G, q)| ≤ N. Thus (4) is proved. Assertion (1) follows from the observation that, if ϕ : G → H is an isogeny over Fq , then by Lemma 3.8 we may arrange the labelling so that I(G, q) ⊂ I(H, q). Assertion (2) holds by virtue of Proposition 3.9.  24 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL 3.1.4. Proofs of Theorem 3.1 and Corollary 3.4. Proof of Theorem 3.1. Let G be a connected semi-simple algebraic group over Fq of Lie type L ∈ LΦ and let G∗ be the dual group. In analysing the right-hand side of (3.2), we deal separately with the sum over central conjugacy classes and the sum over non-central nc conjugacy classes. In analogy with the definition (3.9) of ζG(F (s) we set q) X X c (s) = χ(1)−s , ζG(F q) (g)⊂Z(G∗ (Fq )) χ∈E(G(Fq ),(g)) where the outer sum ranges over the central G∗ -conjugacy classes (g) in G∗ (Fq ) correc nc sponding to semi-simple elements g ∈ Z(G∗ (Fq )). Thus ζG(Fq ) = ζG(F + ζG(F ; cf. (3.2). q) q) c We first consider ζG(Fq ) . There is a constant C0 ∈ R, depending only on Φ, such that |Z(G∗ (Fq ))| ≤ C0 . (3.18) Furthermore, if g ∈ Z(G∗ (Fq )), then the elements of E(G(Fq ), (g)) are in dimensionpreserving bijection with the elements of E(G(Fq ), (1)); see [32, Theorem 5.1]. Thus, (3.19) unip c ζG(F = |Z(G∗ (Fq ))| ζG(F , q) q) and, in combination with (3.18), Proposition 3.5 yields C1 ∈ R and bc (L) ∈ A+ , depending only on Φ respectively L, such that (3.20) unip c ζG(F − |Z(G∗ (Fq ))| = |Z(G∗ (Fq ))| (ζG(F − 1) ∼C0 C1 ξbc (L),q . q) q) We set bc (Φ) = bc (Lsp ). nc . We recall the definition (3.10) of the set bnc (Φ) and set Next we account for ζG(F q) (3.21) a(Φ) = bc (Φ) + bnc (Φ), a(G, q) = bc (L) + bnc (G, q), where bnc (G, q) is defined as in Lemma 3.13. Furthermore, we choose m, depending only on Φ, in accordance with Proposition 3.9. Then the following weaker variants of assertions (1) and (2) of the theorem are clearly satisfied: (1)′ a(G, q) ⊂ a(Φ), (2)′ a(G, q) = a(Φ) whenever G is split and q ≡m 1. Moreover, setting C2 to be the sum of the constant C0 C1 from (3.20) and the constant that Lemma 3.13 supplies, we obtain (3.22) ζG(Fq ) (s) − |Z(G∗ (Fq ))| ∼C2 ξa(G,q),q (s). We now prove that, for q > C2 , |G(Fq )/[G(Fq ), G(Fq )]| = |Z(G∗ (Fq ))|. Indeed, put δ(q) = |G(Fq )/[G(Fq ), G(Fq )]| − |Z(G∗ (Fq ))|. Fix q > C2 and let s = σ ∈ R tend to ∞ in (3.22). The limit of the left-hand side of (3.22) as s = σ → ∞ is equal to δ(q), and q > C2 implies that the limit of C21+σ ξa(G,q) (σ) as σ → ∞ is equal to 0. Thus δ(q) ≤ 0. On the other hand, the limit of the right-hand side of (3.22) as s = σ → ∞ is ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 25 equal to 0, and the limit of C21+σ (ζG(Fq ) (σ) − |Z(G∗ (Fq ))|) as σ → ∞ is non-negative only if δ(q) ≥ 0. Hence δ(q) = 0. Next we show that in the approximations that we seek it is not necessary to distinguish between different members of the isogeny class of G, as we have done up to this point. We define a(L, q) = a(Gad , q) = bc (L) + bnc (Gad , q), where Gad denotes the adjoint quotient of G and claim that assertions (1), (2), (3) in the theorem hold. The statements (1), (2) are clearly special cases of (1)′ , (2)′ . In order to establish (3) we observe that Lemma 3.13 already gives ξa(Gsc ,q),q .1 ξa(G,q),q .1 ξa(Gad ,q),q = ξa(L,q),q . sc where G denotes the simply connected member in the isogeny class of G. Thus it suffices to show that there is a constant C3 ∈ R, depending only on Φ, such that ζGsc (Fq ) − |Gsc (Fq )/[Gsc (Fq ), Gsc (Fq )]| &C3 ζGad (Fq ) − |Gad (Fq )/[Gad (Fq ), Gad (Fq )]|. Write G = Gad (Fq ). Let H denote the image of Gsc (Fq ) under the natural homomorphism Gsc (Fq ) → Gad (Fq ). Then |G| = |Gsc (Fq )|, thus |G : H| = |Z(Gsc (Fq ))| is bounded by C3 = |Z(Gsc )|, depending only on Φ. From (3.22) we observe that the smallest degree of a non-linear character of G is given by an increasing function in q. Taking as a normal subgroup the trivial subgroup, we apply part (3) of Lemma 2.11 to deduce that ζGsc (Fq ) − |Gsc (Fq )/[Gsc (Fq ), Gsc (Fq )]| &1 ζH − |H/[H, H]| &C3 ζG − |G/[G, G]|. Finally, it remains to deduce the estimate (3.1). We need to show that for some constant C4 ∈ R, depending only on Φ, ζG(Fq ) (s) ∼C4 1 + q rk Φ−|Φ + |s . By part (4) of Lemma 3.13, there is a constant C5 ∈ R such that (3.23) 1 + ξbnc (G,q),q (s) ∼C5 1 + q rk Φ−|Φ + |s . Furthermore, using equation (3.19), part (2) of Proposition 3.5, part (3) of Lemma 3.13 and equations (3.18) and (3.23), we conclude that there is a constant C6 ∈ R, depending only on Φ, such that unip c nc nc ζG(Fq ) (s) = ζG(F (s) + ζG(F (s) = |Z(G∗ (Fq ))|ζG(F (s) + ζG(F (s) q) q) q) q) ∼C6 |Z(G∗ (Fq ))| − 1 + 1 + ξbnc (G,q),q (s) ∼C0 +C5 1 + q rk Φ−|Φ proving the claim. This concludes the proof of Theorem 3.1. + |s ,  Proof of Corollary 3.4. We use the inequality 2− dim H q dim H ≤ |H(Fq )| ≤ 2dim H q dim H true for every connected algebraic group H over a finite field Fq ; cf. [35, Lemma 3.5]. Let C1 be the constant from Theorem 3.1. The derived subgroup G′ is connected, semi-simple, 26 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL and has absolute root system Φ. The quotient G/G′ is a torus of dimension dim Z(G). By the Lang–Steinberg theorem [13, Theorem 3.10], there is a short exact sequence 1 → G′ (Fq ) → G(Fq ) → G/G′ (Fq ) → 1, which implies that ζG(Fq ) (s) .1 |G/G′ (Fq )|ζG′ (Fq ) (s) .2dim Z(G) q dim Z(G) ζG′ (Fq ) .C1 q dim Z(G) (1 + q rk Φ−|Φ + |s ). For the opposite inequality, choose a maximal Fq -rational torus T ⊂ G/Z(G) and e B e ⊂ G be the pre-images of a Borel subgroup B ⊂ G/Z(G) containing T. Let T, T, B under the quotient map. By the proof of [6, Lemma 8.4.2], there is C2 ∈ R, dee q ) for which the parabolic pending only on Φ, such that the set of characters ϑ of T(F G(Fq ) e q )|. Thus, we get at least induction IndB(F (ϑ) is irreducible has size at least C2 |T(F e ) q C2 · 2 e dim T e − dim T q e = C2 · 2− dim T q dim Z(G)+rk Φ irreducible representations of dimension e q )| ≤ 2dim G+dim Be q dim G−dim Be = 2dim G+dim Be q |Φ+ | . |G(Fq )|/|B(F Finally, G(Fq ) has |G/G′ (Fq )| ≥ 2− dim Z(G) q dim Z(G) one-dimensional representations factoring through G/G′ (Fq ). Combining these two classes of representations, we get that + ζG(Fq ) (s) &C3 q dim Z(G) (1 + q rk Φ−|Φ |s ) for a suitable C3 ∈ R.  3.2. Applications to Finite Quotients of Arithmetic Groups. In order to apply Theorem 3.1 in a global context, we recall the definition of a Chebotarev set. Definition 3.14. Let K be a number field with ring of integers O, and let ̺ : GalK → G be a continuous homomorphism from the absolute Galois group of K into a finite group G. Let P (̺) denote the set of primes p ∈ Spec(O) such that p is unramified in the extension ker ̺ and the Frobenius conjugacy class (Frobp ) ⊂ GalK associated to p lies in the K⊂K kernel of ̺. A set P ⊂ Spec(O) is a Chebotarev set if it almost contains a set of the form P (̺), in the sense that P (̺) r P is finite. The intersection of two Chebotarev sets is a Chebotarev set. Furthermore, by Chebotarev’s Density Theorem, every Chebotarev set has positive analytic density. We record the following corollary of Theorem 3.1; compare the definition of a(Φ) ∈ A+ in (3.21). Corollary 3.15. Let Φ be a non-trivial root system, and let C ∈ R and a(Φ) ∈ A+ be as in Theorem 3.1. Let K be a number field with ring of integers O. For every affine group scheme G over O whose generic fiber is connected semi-simple with absolute root system Φ, the set of primes p ∈ Spec(O) such that ζG(O/p) − |G(O/p)/[G(O/p), G(O/p)]| ∼C ξa(Φ),|O/p| is a Chebotarev set. Proof. Let G be as in the corollary, and m ∈ N as in Theorem 3.1. The set of primes p such that the reduction of G modulo p is connected, semi-simple, and has absolute root system Φ is cofinite. If M ⊃ K is a splitting field of G and p splits completely in M, then the reduction of G modulo p is split. Hence, there exists a Chebotarev set P ⊂ Spec(O) such that, for all p ∈ P , ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 27 (1) The reduction of G modulo p is connected, split, semi-simple, and has absolute root system Φ. (2) The field O/p contains primitive mth roots of unity, that is |O/p| ≡m 1. Let p ∈ P and set q = |O/p|. By property (1), the group G(O/p) has Lie type L = Lsp = (Φ, Id). Property (2) together with parts (3) and (2) of Theorem 3.1 yields ζG(O/p) − |G(O/p)/[G(O/p), G(O/p)]| ∼C ξa(Φ),q .  Furthermore, an argument similar to the one used in the proof of Theorem 1.5, now based on Theorem 3.1 and Corollary 3.15 instead of Theorem 2.8, gives the following. Corollary 3.16. Let K ⊂ L be number fields with rings of integers OK ⊂ OL , and let G be an affine group scheme defined Q over OK whose generic fiber Q is connected and simply connected semi-simple. Then α( p∈Spec(OK ) G(OK /p)) = α( q∈Spec(OL ) G(OL /q)). We Q remark that one can use Deligne-Lusztig theory to pin down the precise value of α( p G(O/p)) as in Corollary 3.16. For instance, if G is simple of type Aℓ , then the Q abscissa of convergence for the product p G(O/p) of finite groups of Lie type is equal to 2/ℓ and, in particular, tends to 0 as ℓ → ∞. This behavior stands in contrast to the fact the abscissa of convergence α(G(O)) is known to be bounded away from 0; cf. [28, Theorem 8.1]. This underlines that the investigation of α(Γ) for arithmetic groups Γ = G(OS ) requires a more careful analysis. 4. Relative Zeta Functions, Kirillov Orbit Method and Model Theoretic Background 4.1. Relative Zeta Functions and Cohomology. Throughout this section, let G be a group such that Rn (G) is finite for all n ∈ N. Let N ⊂ G be a normal subgroup, and let ϑ be an irreducible finite-dimensional complex representation of N. Recall from Definition 2.10 that Irr(G|ϑ) denotes the set of (equivalence classes of) finite-dimensional irreducible complex representations ̺ of G such that ϑ is a constituent of ResG N ̺, the no
P dim ̺ −s tion of the relative zeta function ζG|ϑ (s) = ̺∈Irr(G|ϑ) dim ϑ , and the notation Rn (G|ϑ) for the number of representations ̺ ∈ Irr(G|ϑ) such that dim ̺ ≤ n dim ϑ. Suppose further that, up to equivalence, ϑ is G-invariant. It is typically not true that the relative zeta function ζG|ϑ is equal to the zeta function of the quotient G/N; for instance, if G is non-abelian, step-2 nilpotent, N = [G, G] and ϑ is non-trivial, then ζG|ϑ is not equal to ζG/N . We describe now a situation in which the two zeta functions are equal. Recall that ϑ defines an element in the second cohomology group H 2 (G/N, C× ) of G/N with values in C× , also known as the Schur multiplier of G/N. The construction is as follows; see [22, Chapter 11]. Suppose ϑ : N → GLd (C). Pick a coset representative e a∈G for every element a of G/N such that e 1 = 1. For every a ∈ G/N, the representations ϑ and ϑea are equivalent, and we choose Ta ∈ GLd (C) such that Ta ϑTa−1 = ϑea ; for T1 28 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL we choose the identity. Then one checks that, for all a1 , a2 ∈ G/N, the transformation −1 Ta−1 T T ϑ(ae1 ae2 (ag 1 a2 ) ) commutes with each element of ϑ(N) and thus defines a scalar 1 a2 a1 a2 β(a1 , a2 ) ∈ C× . The map β : G/N ×G/N → C× is a 2-cocycle representing the cohomology class associated to (G, N, ϑ), which is independent of the choices involved. Lemma 4.1. Let G be a profinite group with an open normal subgroup N ⊳ G, and let ϑ ∈ Irr(N) be a complex representation of N which is G-invariant up to equivalence. If the cohomology class in H 2 (G/N, C× ) associated to (G, N, ϑ) vanishes, then ζG|ϑ = ζG/N . Proof. By [22, Theorem 11.7], the vanishing of the cohomology class implies that ϑ can be extended to a representation ϑe of G. By [22, Theorem 6.16], the map Irr(G/N) → Irr(G|ϑ) given by τ 7→ τ ⊗ ϑe is a bijection, and the claim of the lemma follows.  Lemma 4.2. Let p be a prime number. Let G be a profinite group with an open normal pro-p subgroup N ⊳ G, and let ϑ ∈ Irr(N) be G-invariant up to equivalence. Then the cohomology class in H 2 (G/N, C× ) associated to (G, N, ϑ) has order a power of p. Proof. The dimension of ϑ is a power of p and, for every h ∈ N, the scalar det(ϑ(h)) is a pn th root of unity, for some n. For all a1 , a2 ∈ G/N, taking determinants in the definition −1 of the cocycle β, we get β(a1 , a2 )dim ϑ = det(Ta−1 T T ϑ(ae1 ae2 (ag 1 a2 ) )). Since we are 1 a2 a1 a2 free to arrange det(Tx ) = 1 for x ∈ G/N, we get that β(a, b) is a root of unity of order a power of p.  Lemma 4.3. Let ℓ be a prime number. Suppose that G is a finite group and that N ⊂ G is a central subgroup such that gcd(|N|, ℓ) = 1. Then gcd(|H 2(G, C× )|, ℓ) = 1 if and only if gcd(|H 2 (G/N, C× )|, ℓ) = 1. Proof. Let (Enp,q ) be the Lyndon–Hochschild–Serre spectral sequence associated to the central extension 1 → N → G → G/N → 1. Since the order of N is prime to ℓ, so are the orders of H 1 (N, C× ) and H 2 (N, C× ). Therefore, the orders of E20,2 = H 0 (G/N, H 2(N, C× )) = H 2 (N, C× ) and E21,1 = H 1 (G/N, H 1(N, C× )) = Hom(G/N, H 1(N, C× )) 0,2 1,1 p,q are prime to ℓ and, hence, so are the orders of E∞ and E∞ . The fact that E∞ converges ∗ × 2 × 0,2 1,1 2,0 2 × to H (G, C ) implies that |H (G, C )| = |E∞ | |E∞ | |E∞ |, so |H (G, C )| is prime to ℓ 2,0 if and only if |E∞ | is prime to ℓ. Since the order of E20,1 = H 0 (G/N, H 1(N, C× )) = H 1 (N, C× ) is prime to ℓ, we get that 2,0 the order of E∞ = E32,0 is prime to ℓ if and only if the order of E22,0 = H 2 (G/N, C× ) is prime to ℓ, yielding the result.  Lemma 4.4. For every root system Φ there is a constant C ∈ R such that, for every finite field Fq of characteristic greater than C and for every connected reductive Fq -algebraic group G with absolute root system Φ, the size of H 2 (G(Fq ), C× ) is prime to q. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 29 Proof. Let Fq be a finite field of characteristic p and let G be a connected reductive Fq -algebraic group with absolute root system Φ. Assume first that G is semi-simple. Then there are almost simple groups G1 , . . . , Gn such that G is a quotient of G1 × · · · × Gn by a central subgroup Z, and both n and the ranks of the groups Gi are bounded in terms of Φ. In particular, the size of Z is bounded in terms of Φ. From the exact sequence 0 → Z(Fq ) → Yn i=1 Gi (Fq ) → G(Fq ) → H 1 (GalFq , Z) Q we conclude that both the kernel and the cokernel of the map ni=1 Gi (Fq ) → G(Fq ) have sizes bounded in terms of Φ. It is known that the sizes of H 1 (Gi (Fq ), C× ) and H 2 (Gi (Fq ), C× ) are bounded in terms Q of Φ; see, for example,Q[11, Table 5]. By the Künneth formula, the sizes of H 1 ( ni=1 Gi (Fq ), C× ) and H 2 ( ni=1 Gi (Fq ), C× ) are bounded C1 ∈ R. Qn by some constant × 2 In particular, if p is greater than C1 , then the size of H ( i=1 Gi (Fq ), C ) is prime to q. 2 × By Lemma 4.3, the same is true for the size of H Qn(G(Fq ), C ) if p is larger than the size of the kernel and cokernel of the quotient map i=1 Gi (Fq ) → G(Fq ). Now assume that G is merely reductive. Let S = [G, G] be the derived subgroup of G and let T = Z(G)◦ . Then T is a torus, S is semi-simple, and G(Fq ) is a quotient of T(Fq ) × S(Fq ) by a central subgroup, whose size is bounded in terms of Φ. As shown above, if p is sufficiently large, then the size of H 2 (S(Fq ), C× ) is prime to q; a similar claim for H 1 (S(Fq ), C× ) also holds. Since the size of T(Fq ) is prime to q, so are the orders of its first and second cohomology groups. By the Künneth formula, the size of H 2 (T(Fq ) × S(Fq ), C× ) is prime to q. By Lemma 4.3, if we assume, in addition, that p is larger than the size of the kernel of T(Fq )×S(Fq ) → G(Fq ), then the size of H 2 (G(Fq ), C× ) is prime to q.  4.2. Kirillov Orbit Method. All pro-p groups in this section are open subgroups of G(OL,q ), where G is an affine group scheme over the ring of integers OL of a number field L and OL,q is the completion of OL at a prime q lying above a rational prime p. We fix an embedding G ⊂ GLN , for a suitable N ∈ N, and denote by g ⊂ glN the Lie algebra of G. Definition 4.5. Let q be a prime of OL . We say that a pro-p subgroup H ⊂ G(OL,q ) is good if the following two conditions hold. P n n−1 (X−1) (1) The logarithm series log(X) = ∞ converges on H, setting up an n=1 (−1) n injective map log : H → log H ⊂ g(O ), and the exponential series exp(X) = L,q P∞ X n n=0 n! converges on log H, yielding the inverse map exp : log H → H. (2) The image log H is closed under addition and the Lie bracket, thus forming a Zp -Lie lattice. It is also closed under the adjoint action of H. For all A, B ∈ log H, the 30 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Hausdorff formula (e.g., see [23, Chapter V]) holds: P ∞ −1 X ( m (−1)m X i=1 (ri + si )) log(exp(A) · exp(B)) = Rr ,s ,...,r ,s (A, B), m r +s >0 r1 ! · s1 ! · . . . · rm ! · sm ! 1 1 m m m=1 i i where the Lie polynomials Rr1 ,s1 ,...,rm ,sm (A, B) are defined  r s rm  ad(A) 1 ad(B) 1 · · · ad(A) (B) Rr1 ,s1 ,...,rm ,sm (A, B) = ad(A)r1 ad(B)s1 · · · ad(B)rm−1 (A)  0 by if sm = 1, if rm = 1, sm = 0, otherwise. Lemma 4.6. Let G ⊂ GLN be as above and let q be a prime of OL extending a rational prime p > [L : Q]N 2 . Then every pro-p subgroup H ⊂ G(OL,q ) is good. Proof. Pro-p groups which are saturable in the sense of Lazard – for recent characterizations see [26, 17] – are good in the sense of Definition 4.5. The assertion thus follows from [26, Corollary 1.5] which implies that every pro-p subgroup H ⊂ GLN (OL,q ) is saturable.  A good pro-p group H acts on h = log H via the adjoint action Ad : H → Aut(h). We denote by Ad(h)(A) the image of A ∈ h under the adjoint action of h ∈ H. The adjoint action induces the co-adjoint action of H on the Pontryagin dual h∨ = Homcont (h, C× ), consisting of all continuous homomorphisms from the abelian pro-p group h to C× . Concretely, for h ∈ H, A ∈ h, and ϑ : h → C× , one defines
(Ad∗ (h)(ϑ)) (A) = ϑ Ad(h−1 )(A) . Equivalence classes of irreducible representations of H are parametrized by the corresponding characters. Accordingly, we use the notation Irr(H) in a flexible way to denote also the set of irreducible complex characters of H. The Kirillov orbit method for p-adic analytic pro-p groups yields the following description of the irreducible characters of H. Proposition 4.7. Let G ⊂ GLN be as above. For almost all primes q of OL , every pro-p subgroup H ⊂ G(OL,q ) is good and, setting h = log H, the following hold. (1) There is a function Ω : h∨ → Irr(H) which is constant on co-adjoint orbits and induces a bijection between the set of co-adjoint orbits in h∨ and the set of irreducible characters of H. (2) For ϑ ∈ h∨ , the character Ω(ϑ) is given by X 1 Ω(ϑ)(h) = ϕ(log(h)) (h ∈ H). |Ad∗ (H)(ϑ)|1/2 ∗ ϕ∈Ad (H)(ϑ) In particular, the degree of Ω(ϑ) is dim Ω(ϑ) = |Ad∗ (H)(ϑ)|1/2 . (3) Every g ∈ G(OL,q ) that normalizes the subgroup H also normalizes the Lie lattice h and Ω(ϑ) g = Ω(Ad∗ (g −1)(ϑ)) for ϑ ∈ h∨ . ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 31 Proof. Let S be the finite set of primes q of OL that extend a rational prime p with p ≤ [L : Q]N 2 . The assumptions then imply that H ⊂ G(OL,q ) is saturable for q ∈ / S (see [17, Theorem A]), in particular good in the sense of Definition 4.5. As saturable pro-p groups of dimension at most p are potent, the assertions follow from [16, Theorem 5.2].  We refer to the map Ω in Proposition 4.7 as the orbit method map. Lemma 4.8. Let G ⊂ GLN be as above, and let G be an open subgroup of G(OL,q ) for some prime q of OL , with open normal subgroups K ⊂ H ⊂ G. Suppose that H and K are good pro-p groups, with Lie lattices h = log H and k = log K, and that the irreducible characters of H and K are described by the orbit method as in Proposition 4.7. Then ζG (s) = X 1 (dim Ω(ϑ))−s ζG|Ω(ϑ) (s). |Ad (G)(ϑ)| ∗ ϑ∈h∨ Furthermore, if τ ∈ k∨ , then ζG|Ω(τ ) (s) = X ϑ∈h∨ , ϑ|k =τ |Ad∗ (H)(τ )| |Ad∗ (G)(ϑ)|  dim Ω(ϑ) dim Ω(τ ) −s ζG|Ω(ϑ) (s). Proof. For each χ ∈ Irr(H), the set Irr(G|χ) depends only on the G-orbit χG . Moreover, the sets Irr(G|χ), indexed by the orbits χG of irreducible characters of H, form a partition of Irr(G). Choosing representatives χi , i ∈ I, for the G-orbits, we get X X ζG (s) = (dim χi )−s ζG|χi (s) = |χG |−1 (dim χ)−s ζG|χ (s). i∈I χ∈Irr(H) Consider the orbit method map Ω : h∨ → Irr(H). Its fibers are the H-coadjoint orbits. Moreover, Ω is G-equivariant, so the pre-images of the G-orbits in Irr(H) are the G-orbits in h∨ . Hence, X ζG (s) = |χG |−1 (dim χ)−s ζG|χ (s) χ∈Irr(H) = X ϑ∈h∨ = X ϑ∈h∨ 1 (dim Ω(ϑ))−s ζG|Ω(ϑ) (s) G |Ad (H)(ϑ)| · |Ω(ϑ) | ∗ 1 (dim Ω(ϑ))−s ζG|Ω(ϑ) (s). |Ad (G)(ϑ)| ∗ The proof of the second statement is similar, using the following consequence of Proposition 4.7: for τ ∈ k∨ and ϑ ∈ h∨ , the character Ω(τ ) is a constituent of the restriction of Ω(ϑ) to K if and only if ϑ|k = τ h for a suitable h ∈ H.  32 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL 4.3. Quantifier-Free Definable Sets and Functions. We use several notions from model theory, which we summarize below. For more details we refer to [7]. Fix a firstorder language and a theory T, that is a consistent set of sentences, in that language. Let (Models T ) be the category whose objects are models of T and whose morphisms are elementary embeddings. Let x = (x1 , . . . , xn ) denote an n-tuple of variables. A formula ϕ(x) gives rise to a functor X = Xϕ : (Models T ) → (Sets) that sends a model M of T to the set X(M) = {(a1 , . . . , an ) ∈ M n | the sentence ϕ(a1 , . . . , an ) holds in M } . We call such a functor X a definable functor or a definable set in T, and denote Xϕ also by {x | ϕ(x)}. By Gödel’s completeness theorem, the functors associated to formulae ϕ and ψ are equal if and only if the sentence ∀x (ϕ(x) ↔ ψ(x)) can be proved from T. We say that a definable set is quantifier-free definable if it is associated to some quantifier-free formula. For a definable set X arising from a formula ϕ, the notation a ∈ X is employed in two different contexts: if a ∈ M n for some model M of T it means ϕ(a) holds in M, whereas if a is a tuple of terms in the underlying language it simply stands for ϕ(a). The usual pointwise operators ∩, ∪, and × on functors to sets take definable sets to definable sets. If X and Y are definable sets, we write X ⊂ Y if X(M) ⊂ Y(M) for all models M of T. If X and Y are associated to the formulae ϕ and ψ, then X ⊂ Y if and only if the sentence ∀x (ϕ(x) → ψ(x)) can be proved from T. Example 4.9. We consider the first-order language of rings and the theory T fields of fields. (1) For every n ∈ N, the formula 0 = 0 (in variables x1 , . . . , xn ) yields a definable set X with X(F ) = F n for every field F . The associated definable set is called n-dimensional affine space, and denoted by An . (2) More generally, every affine scheme X over Z can be considered as a quantifier-free definable set. This means that there is a quantifier-free definable set X such that X(F ) = X(F ) for every field F . (3) The definable set Y arising from the formula ∃y (y 2 = x) is not quantifier-free. Indeed, every quantifier-free definable set in the theory of fields is a Boolean combination of affine varieties. It follows that if X ⊂ A1 is quantifier-free, then there is a constant C ∈ R such that |X(Fp )| or |Fp r X(Fp )| is bounded by C, uniformly for all primes p. However, |Y(Fp )| = (p + 1)/2 if p > 2. The dimension of a definable set X ⊂ An is the dimension of its Zariski closure in An ; cf. [10, Section 3]. Let X and Y be definable sets in a theory T. A natural transformation f : X → Y is called a definable function if the functor sending M ∈ (Models T ) to the graph of the map f (M) is definable. This means that the graph of f , considered as a functor, is definable. On some occasions, if f : X → Y is a definable function, we say that X is a definable family of (definable) sets with base Y. For y ∈ Y, we denote the fiber f −1 (y) by Xy . It is a definable set in the enriched language obtained by adding the coordinates of y as constants. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 33 Throughout the remainder of the section, let R denote a commutative unital ring. Definition 4.10. Consider the language of rings enriched by constant symbols ca for all a ∈ R. Let T fields,R be the theory consisting of the axioms of fields and the statements ca · cb = cab , ca + cb = ca+b for all a, b ∈ R. A model for T fields,R is a field F together with a ring homomorphism R → F , which we routinely omit from the notation. As in Example 4.9, every affine scheme over R gives rise to a quantifier-free definable set in T fields,R . We explain now how projective spaces arise as definable sets in the theory T fields,R . Let g be a free R-module of finite rank n with an R-basis e1 , . . . , en . We view g as a definable set in T fields,R via g(F ) = g ⊗R F for F ∈ (Models T fields,R ). The chosen basis allows us to identify g with An . The definable set P = {(1)} × An−1 ⊔ {(0, 1)} × An−2 ⊔ . . . ⊔ {(0, 0, . . . , 1)} × A0 ⊂ An plays the role of projective space over g in the category of definable sets, in the following sense. Consider the definable family V ⊂ P × g given by the condition that a a ((a1 , . . . , an ), (v1 , . . . , vn )) ∈ P × g is in V if and only if all minors i j , 1 ≤ i < j ≤ n, vi vj vanish. For every definable set S and every definable set X ⊂ S×g with the property that, for every s ∈ S, the fiber Xs is a line in g – that is, a one-dimensional linear subfunctor of g – there is a unique definable map f : S → P such that the pull-back f ∗ V ⊂ S × g of V via f is equal to X. Choosing a different (definable) basis for g, we obtain a different universal family over P, but there is a quantifier-free definable map from P to itself that interchanges the two universal families. More generally, for 0 ≤ d ≤ n, there is a quantifier-free definable set that functions as the Grassmannian of d-dimensional subspaces in g ⊗R F for every model F of T fields,R . The union of these, over all dimensions d, is the Grassmannian Gr(g) of g. Definition 4.11. Let g ⊂ glN (R) be a free R-module which is closed under Lie brackets. In the theory T fields,R , let GrLie (g) be the subfunctor of Gr(g) given by F 7→ {Lie subalgebras of g ⊗R F } , and let Grnilp Lie (g) be the subfunctor of GrLie (g) given by F 7→ {Lie subalgebras of g ⊗R F consisting of nilpotent matrices} . Proposition 4.12. Let R be a commutative unital ring, and let g ⊂ glN (R) be a free R-module which is closed under Lie brackets. View g as a definable set in T fields,R via g(F ) = g ⊗R F for F ∈ (Models T fields,R ). Then the following hold. (1) GrLie (g) and Grnilp Lie (g) are quantifier-free definable subfunctors of Gr(g). (2) The natural transformation GrLie (g) → Grnilp Lie (g) induced by taking a Lie algebra h ⊂ g(F ) to the subalgebra of nilpotent matrices in the solvable radical Rad(h) of h is quantifier-free definable. 34 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL (3) The natural transformation Grnilp Lie (g) → GrLie (g) induced by taking a Lie algebra h ⊂ g(F ) consisting of nilpotent matrices to its normalizer Ng(F ) (h) in g(F ) is quantifierfree definable. (4) There are a constant p0 ∈ N and a quantifier-free definable function from GrLie (g) to the set of all root systems of rank at most N such that the following is true: for every Lie algebra h ⊂ g(F ) with F satisfying char(F ) = 0 or char(F ) ≥ p0 , the Lie algebra h/ Rad(h) is semi-simple of classical type and the value of the function at h is the absolute root system of h/ Rad(h). (5) Suppose that g is the Lie algebra of an affine group scheme G over R. Then there is a quantifier-free definable subset of G × GrLie (g) whose fiber over a Lie algebra h ⊂ g(F ) is the normalizer NG(F ) (h) of h in G(F ). We use the following well-known characterization of quantifier-free definable sets; e.g., see [25, Theorem 8.11]. Lemma 4.13. Let R be a commutative unital ring, and let X ⊂ An be a definable set in the theory T fields,R . Then the following are equivalent. (1) For every two models F ⊂ E of T fields,R , we have X(F ) = X(E) ∩ F n . (2) For every model F of T fields,R , we have X(F ) = X(F alg ) ∩ F n . (3) X is quantifier-free definable. Proof. The implications (1) ⇒ (2) and (3) ⇒ (1) are clear. We prove (2) ⇒ (3). By elimination of quantifiers over algebraically closed fields with coefficients in R, there is a quantifier-free definable set Y such that X(E) = Y(E) for every algebraically closed model E of T fields,R . Let F be a model of T fields,R . By (2) and the implication (3) ⇒ (1), applied to Y, we obtain X(F ) = X(F alg ) ∩ F n = Y(F alg ) ∩ F n = Y(F ), so X = Y.  Remark 4.14. We also use variants of Lemma 4.13 which characterize quantifier-free definable sets in (i) the theory T perf.-fields,p,R of perfect fields of characteristic p together with a homomorphism from R and (ii) the theory T Hen,K,0 of Henselian valued fields of residue characteristic 0 together with a homomorphism from field K; cf. Definition 4.17. The proof proceeds in the same way. Proof of Proposition 4.12. We show in each case that the functor or the graph of the natural transformation in question is definable. In some cases, the formulae that we supply are polynomial, and hence quantifier-free. In the remaining cases, we explain how to apply Lemma 4.13 in order to obtain that the functor or graph in question is actually quantifier-free definable. Put n = dim g and fix an R-basis e1 , . . . , en of g. There are a finite Zariski open affine cover (Uι )ι∈I of Gr(g), non-negative integers (dι )ι∈I , and regular functions xι,1 , . . . , xι,n : Uι → g, for ι ∈ I, such that for each model F of T fields,R , ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 35 ◦ the dimension of every h ∈ Uι (F ) is equal to dι , ◦ for every h ∈ Uι (F ), the elements xι,1 (h), . . . , xι,dι (h) yield a linear basis for h and the elements xι,1 (h), . . . , xι,n (h) yield a linear basis for g(F ). In order to prove that a subfunctor X ⊂ Gr(g) is definable, it is enough to show that X ∩ Uι is definable for every ι ∈ I. A similar claim holds for natural transformations. Fix therefore ι, ι′ ∈ I and write U = Uι , U′ = Uι′ , d = d ι , d ′ = d ι′ , and xi = xι,i , x′i = xι′ ,i for i ∈ {1, . . . , n}.

(1) Consider the d+1 × n -matrix A whose first d rows record the coordinates of the 2
functions x1 , . . . , xd and whose last d2 rows record the coordinates of the Lie brackets [xi , xj ], 1 ≤ i < j ≤ a, all with respect to the basis e1 , . . . , en . The functor GrLie (g) ∩ U is definable by the polynomial condition rk A = d. The functor Grnilp Lie (g) ∩ U is definable by the conjunction of the previous condition and the polynomial condition that all products of the xi of length N vanish, i.e., that Q N N j=1 xij = 0 for all (i1 , . . . , iN ) ∈ {1, . . . , n} . (2) Let F be a model of T fields,R and h ∈ (GrLie (g) ∩ U)(F ). Recall that an element X ∈ h is in the solvable radical Rad(h) of h if and only if the Lie ideal [X, h] generated by X in h is solvable. As an F -vector space, [X, h] is spanned by the set S consisting of all elements [X, xi1 (h), . . . , xid−1 (h)], where i1 , . . . , id−1 ∈ {1, . . . , d}. The Lie words wi , i ∈ N, defining the terms of the derived series are w1 (z1 , z2 ) = [z1 , z2 ] and wi (z1 , . . . , z2i ) = [wi−1 (z1 , . . . , z2i−1 ), wi−1 (z2i−1 +1 , . . . , z2i )] for i ≥ 2. By linearity, [X, h] is solvable if and only if wn (Z1 , . . . , Z2n ) = 0 for all Z1 , . . . , Z2n ∈ S. Using (1), we deduce that the functor (4.1) F 7→ {(h, X) ∈ (GrLie (g) ∩ U)(F ) × g(F ) | X ∈ Rad(h)} is quantifier-free definable. Clearly, we can express nilpotency of an element X ∈ h by a polynomial formula. Consequently, also the functor F 7→ {(h, X) ∈ (GrLie (g) ∩ U)(F ) × g(F ) | X ∈ Rad(h) is nilpotent} is quantifier-free definable. Using quantifiers, we deduce that the functor   (h, k) ∈ (GrLie (g) ∩ U)(F ) × (Grnilp (g) ∩ U′ )(F ) | Lie (4.2) F 7→ k is the collection of nilpotent elements in Rad(h) is definable and thus the graph of a natural transformation, namely the one we are interested in. It remains to prove that the functor (4.2) is quantifier-free definable. By Lemma 4.13, it suffices to consider the following. Let F ⊂ E be models of T fields,R , and for subalgebras h, k ⊂ g(F ) write hE = h⊗F E and kE = k ⊗F E. We claim: if kE consists of the nilpotent elements of Rad(hE ), then k consists of the nilpotent elements of Rad(h). 36 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Clearly, X ∈ h is nilpotent as an element of hE if and only if it is nilpotent as an element of h. Moreover, because the functor (4.1) is quantifier-free definable, we have Rad(hE ) ∩ g(F ) = Rad(h). This proves the claim. (3) We first show that the functor N given by  F 7→ (h, k) ∈ Gr(g)(F )2 | ∀Y ∈ g(F ) : (Y ∈ k ↔ (∀X ∈ h : [X, Y ] ∈ h)) is definable. As before, it is enough to prove this Zariski locally around fixed elements h ∈ U(F ) and k ∈ U′ (F ), of dimensions d and d′ . The intersection N ∩ (U × U′ ) is given by the formula (4.3) ∀a1 , . . . , an h Xn ^d ∃b1 , . . . , bd xi , j=1 i=1 aj x′j i = Xd m=1 bm xm  ←→ a d′ +1  = . . . = an = 0 . We now show that this formula is equivalent to a quantifier-free formula. To this end, consider first the (d(d′ + 1) × n)-matrix A whose first d rows record the coordinates of x1 , . . . , xd and whose last dd′ rows record the coordinates of the Lie brackets [xi , x′j ], where 1 ≤ i ≤ d and 1 ≤ j ≤ d′ , all with respect to the basis e1 , . . . , en . The polynomial condition rk(A) = d ensures that hx′1 , . . . , x′d′ i is contained in the Lie normalizer of hx1 . . . , xd i. Consider now, for 1 ≤ i ≤ d and coordinates ad′ +1 , . . . , an , the ((d + 1) × n)-matrix Bi (ad′ +1 , . . . , an ) whose first d rowsPrecord the coordinates of x1 , . . . , xd and whose last row records the coordinates of [xi , nj=d′ +1 aj x′j ]. Formula (4.3) is equivalent to (rk(A) = d) ∧ ∀ad′ +1 , . . . , an ^d i=1 rk(Bi (ad′ +1 , . . . , an )) = d   ←→ ad′ +1 = · · · = an = 0 . We claim that this is equivalent to a polynomial condition in the coordinate functions x1 , . . . , xd , x′1 , . . . , x′d′ . Indeed, for every i ∈ {1, . . . , d}, the condition rk(Bi (ad′ +1 , . . . , an )) = d is linear in ad′ +1 , . . . , an and polynomial in x1 , . . . , xd , x′1 , . . . , x′d′ . The condition that the resulting system of linear equations for ad′ +1 , . . . , an only has the trivial solution is polynomial in x1 , . . . , xd , x′1 , . . . , x′d′ . (4) The theory of modular Lie algebras, and in particular the classification of semisimple Lie algebras, is rather more involved than in characteristic 0. However, there exists p0 ∈ N such that the classification of semi-simple Lie algebras of dimension at most n over every algebraically closed field of characteristic at least p0 is completely analogous to the well-known classification in characteristic 0: the Lie algebras are of classical type and parametrized by suitable root systems. One can deduce this, for instance, from Robinson’s Principle, according to which a first order statement in the language of fields is true in algebraically closed fields of characteristic 0 if and only if it is true in algebraically closed fields of sufficiently large characteristic. We may thus restrict attention to fields F with char(F ) = 0 or char(F ) ≥ p0 . ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 37 Since the absolute root system of a Lie algebra does not change under extension of scalars, it suffices, by Lemma 4.13, to produce a definable function with the required properties. For every root system Φ, let sΦ be the split Lie algebra, given in terms of a corresponding Chevalley basis, with root system Φ; see [23, Chapter IV]. For every subalgebra h of glN (F ), the algebra h/ Rad(h) is semi-simple and has rank at most N. By adjoining the characteristic roots of basis elements of a Cartan subalgebra of h in the adjoint action, one obtains a splitting field E for h. Clearly, the degree of such a field E over F is bounded in terms of N. Therefore, it suffices to show that, for any given m ∈ N, the functor   h ∈ GrLie (g)(F ) | F 7→ ∃ field ext. F ⊂ E of degree m such that (h/ Rad(h)) ⊗F E ∼ = sΦ (E) is definable. Using Boolean combinations of formulae, it is enough to show that the functor   h ∈ GrLie (g)(F ) | (4.4) F 7→ ∃ field ext. F ⊂ E of degree m and an embedding sΦ (E) → h ⊗F E is definable. It is enough to check definability on an open neighborhood GrLie (g) ∩ U of an element h ∈ GrLie (g)(F ). Field extensions F ⊂ E of degree m can be modelled on the vector space F m via a set of structure constants c = (ckij )m i,j,k=1 in F . The latter supply a binary operation  Xm m F m × F m → F m , (a1 , . . . , am ) ∗ (b1 , . . . , bm ) = . ai bj ckij i,j=1 k=1 The condition that the multiplication ∗, together with vector addition, defines a field extension of F is a first-order condition on c. Furthermore, for every c giving rise to an extension F ⊂ E, an E-linear map T : sΦ (E) → h ⊗F E can be described, locally, by a (dim sΦ × d)-matrix over F m , with respect to the Chevalley basis of sΦ and the basis x1 (h), . . . , xd (h) of h. The condition that the map T is an embedding and preserves Lie brackets is polynomial in the entries of the corresponding (dim sΦ × d × m)-array t over F . Thus the functor (4.4) is indeed definable. (5) Let X be the functor F 7→ {(g, h) ∈ G(F ) × GrLie (g)(F ) | Ad(g)h = h} . The functor X ∩ (G × (GrLie (g) ∩ U))) is definable by the formula  Xd  Xd (4.5) (∀a1 , . . . , ad )(∃b1 , . . . , bd ) Ad(g) aj xj (h) = j=1 k=1  bk xk (h) , where the operation of Ad(g) is given by polynomial expressions involving the entries of the matrix g. To see that the functor is quantifier-free definable, consider the (2d × n)-matrix A = A(g, h), whose first d rows record the coordinates of x1 , . . . , xd and whose last d rows record the coordinates of Ad(g)x1 , . . . , Ad(g)xd , all with respect to the basis e1 , . . . , en . The polynomial condition rk(A(g, h)) = d is equivalent to (4.5).  38 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL The following proposition can be found, for example, in [8, Théorème 6.4] or [9, Main Theorem]. Proposition 4.15. Suppose that ϕ(x, y) is a first-order formula in the language of rings, where x = (x1 , . . . , xm ) and y = (y1 , . . . , yn ). There is a constant C ∈ R such that, for every finite field Fq and every a ∈ Fnq , there is a natural number d such that the size of  −1 d the set x ∈ Fm q and Cq d . q | ϕ(x, a) holds in Fq is either 0, or between C Lemma 4.16. Let G be an affine algebraic group over a finite field Fq with at most C connected components, and let g be the Lie algebra of G. (1) Writing D1 = C 2dim G , the estimates D1−1 |g(Fqn )| < |G(Fqn )| < D1 |g(Fqn )| hold for every finite extension Fq ⊂ Fqn . (2) Suppose that G acts on a variety X in such a way that the stabilizer H of a point x ∈ X(Fqn ) in G has less than C connected components. Writing h for Lie algebra of H and D2 = C 2 2dim G , the following estimates hold: |g(Fqn )| |g(Fqn )| D2−1 ≤ |G(Fqn )x| ≤ D2 . |h(Fqn )| |h(Fqn )| Proof. We use the inequality 2− dim G q n dim G ≤ |G◦ (Fqn )| ≤ 2dim G q n dim G for the connected Fq -algebraic group G◦ ; cf. [35, Lemma 3.5]. Observing that |G◦ (Fqn )| ≤ |G(Fqn )| ≤ |G/G◦ | · |G◦ (Fqn )| and |g(Fqn )| = q n dim G , we deduce (1). Claim (2) follows from (1), by applying the orbit stabilizer theorem.  4.4. Valued Fields. We use the Denef–Pas language of valued fields; see, for example [10, Section 2]. It is a three-sorted, first-order language. The three sorts are the valued field sort F, the residue field sort k, and the value group sort Γ. The function symbols are +val , ×val +res , ×res +gr val ac from from from from from pairs of valued field sort variables to one valued field sort variable, pairs of residue field sort variables to one residue field sort variable, pairs of value group sort variables to one value group sort variable, one valued field sort to one value group sort, one valued field sort to one residue field sort. In addition there is one binary relation symbol, <, between two value group sort variables. For us, the important structures for the language of valued fields come from discrete valuation fields. Given a discrete valuation field F with a uniformizer ̟, we interpret the valued field sort F as F , the residue field sort k as the residue field of F , and the value group sort Γ as the value group of F which we identify with Z. The functions +val , ×val , +res , ×res , +gr , and the relation < are interpreted as the usual operations and order relation. Finally, the function symbol val is interpreted as the valuation map, and the function symbol ac is interpreted as the angular component map ac(x) ≡ x̟ − val(x) (mod ̟) for x ∈ F r {0}. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 39 The values of val(0) and ac(0) are chosen to be ∞ and 0. We will only use theories for which F is a valued field with residue field k and value group Γ. Definable sets and functions are introduced similarly as for languages with only one variable sort. The definable set {x ∈ F | val(x) ≥ 0} is denoted by O. We let red : O → k denote the reduction modulo the maximal ideal, i.e. the definable map ( ac(x), if val(x) = 0, red(x) = 0, if val(x) > 0. When several valuation rings are involved, we sometimes use subscripts to distinguish between the various realizations of the reduction map. We use the function symbol red(·) also to denote the componentwise reduction of a matrix or a tuple. In the latter case we also write red×n to highlight the n-arity. Likewise, we write ac×n (x) and val×n (x) when we apply the map ac or val coordinatewise to a tuple x ∈ Fn . For every discrete valuation field F , the set O(F ) is the valuation ring OF . Every OF -scheme X gives rise to three definable sets in the Denef–Pas language augmented by constants from OF . Indeed, let x = (x1 , . . . , xn ) denote an n-tuple of variables and suppose that X is given as the vanishing set of polynomials f1 (x), . . . , fm (x) ∈ OF [x]. The first definable set is the set XF of all zeros of fi (x), 1 ≤ i ≤ m, in Fn . The second is the set Xk of zeros of the reductions of fi (x), 1 ≤ i ≤ m, in kn . The third is XO = XF ∩ On . For instance, if G is an affine group scheme defined over OF , and F ⊂ E is a finite extension of discrete valuation fields, with ring of integers OE and residue field Fq , then GF (E) = G(E), Gk (E) = G(Fq ), GO (E) = G(OE ). The constructions of Section 4.3 can be applied to definable sets of sorts F and k. For example, let F be a valued field and let g be a Lie algebra scheme over OF . Applying Proposition 4.12 to the quantifier-free definable set gk , we get a quantifier-free definable set GrLie (gk ) such that, for every extension F ⊂ E, the set GrLie (gk )(E) is the collection of all Lie subalgebras of gk (E) = g ⊗OF k(E). The assertions of Proposition 4.12 carry over to analogous statements for definable sets of sorts F and k. Definition 4.17. Let T Hen,0 be the theory of Henselian valued fields of residue characteristic 0, that is the theory generated by the axioms stating that a discrete valuation field is Henselian (i.e., the valuation ring satisfies the conclusions of Hensel’s Lemma), and that its residue characteristic is not equal to p, for every rational prime p. Furthermore, given a field K of characteristic 0, we consider also the Denef–Pas language enriched by constant symbols ca of valued field sort for all a ∈ K and T Hen,K,0 denotes the theory of Henselian valued fields of residue characteristic 0 together with the statements ca ·cb = cab , ca + cb = ca+b for all a, b ∈ K; cf. Definition 4.10. The theory T Hen,0 admits partial elimination of quantifiers; see [36, Theorem 4.1] or [19]. By a standard argument, the same holds for the theory T Hen,K,0 in every extended language, as discussed above. We record this fact as follows. 40 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Theorem 4.18. Every Denef–Pas formula ϕ is T Hen,0 -equivalent to a formula ψ without valued field quantifiers. For every field K of characteristic 0, the analogous statement holds true for T Hen,K,0. Consider a number field L. None of the local fields Lq , where q ranges over the primes of OL , is a model for T Hen,0 . Nevertheless, we will use theorems proved in T Hen,K,0 in the following way. Suppose that X and Y are two definable sets, and that X = Y holds in T Hen,K,0. Then the equivalence of X and Y can be proved by only finitely many axioms of T Hen,K,0. Hence, X(E) = Y(E) is true is true for K ⊂ E, assuming only that the valued field E is Henselian and that the characteristic of the residue field of E is greater than some constant (depending on X and Y). In particular, we deduce that X(Lq ) = Y(Lq ) for almost all primes q of OL . 5. Parametrizing Representations In this section, we consider an affine group scheme G ⊂ GLN over the ring of integers OK of a number field K whose generic fiber is semi-simple. Let g ⊂ glN be the Lie algebra of G. We consider G and g as quantifier-free definable sets over the first-order language of valued fields enriched by adding constant symbols for the elements of K, and work in the theory T Hen,K,0 of Henselian fields over K with residue characteristic 0. Our aim is to prove the following result. Theorem 5.1. Let K be a number field with ring of integers OK , and let G ⊂ GLN be an affine group scheme over OK whose generic fiber is semi-simple. There are a (dim G + 1)dimensional quantifier-free definable set Z ⊂ Odim G+1 , quantifier-free definable functions f1 , f2 : Z → Γ, and a constant C ∈ R such that, for every finite field extension K ⊂ L and almost all primes q of OL , Z ζG(OL,q ) (s) − ζG(OL /q) (s) ∼C |OL /q|f1(z)−f2 (z)s dλ(z), Z (Lq ) dim G+1 where λ is the additive Haar measure on Lqdim G+1 normalized so that λ(OL,q ) = 1. Throughout the proof of Theorem 5.1, there are places where we omit finitely many primes q of OL . Observation 5.3 collects most of the restrictions that we impose. In addition, we exclude in the proof of Proposition 5.9 and all consequences thereof finitely many primes that are not specified explicitly; this is due to partial elimination of quantifiers. In the actual proof of Theorem 5.1 in Section 5.4, an application of Proposition 4.12 also requires us to disregard finitely many primes. Throughout we collectively write “for almost all primes” to refer to these restrictions. The choice of primes we omit may depend on L. However, we emphasize that the definable set Z and the definable functions f1 , f2 in Theorem 5.1 do not depend on the choice of omitted primes. 5.1. Relative Orbit Method. We continue to use the notation set up to formulate Theorem 5.1. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 41 Definition 5.2. Let X be the quantifier-free definable set gO × (O r {0}). Throughout we fix a non-degenerate, invariant and Ad(G)-invariant bilinear form h·, ·i on g, e.g., the Killing form, and we consider finite extensions K ⊂ L. For q ∈ Spec(OL ) lying above a rational prime p, we denote by g(OL,q )∨ the Pontryagin dual of the abelian pro-p group g(OL,q ). Observation 5.3. By omitting finitely many primes of OL , we may concentrate in the proof of Theorem 5.1 on q ∈ Spec(OL ) such that the following conditions hold. ◦ pZ = q ∩ Z is unramified in Q ⊂ L, i.e., the valuation of the integer p in Lq is 1; furthermore, p > N + 1, and p ∤ |H 2(H(Fq ), C× )| for every finite field Fq of characteristic p and every connected reductive Fq -algebraic group H with dim H ≤ dim G; see Lemma 4.4. ◦ The form h·, ·i is non-degenerate on g(OL,q ). ◦ There is a surjective map Πq : X (Lq ) → g(OL,q )∨ , taking the pair x = (Ax , zx ) to the homomorphism of abelian groups    hAx , Bi × Πq (x) : g(OL,q) → C , B 7→ exp 2πi · TrLq |Qp . zx ◦ Every pro-nilpotent Lie subring of g(OL,q ) containing the 1st principal congruence Lie sublattice g(1) (OL,q ) yields a pro-p subgroup of G(OL,q ) via the exponential map, and the Kirillov orbit method applies to pro-p subgroups of G(OL,q ) as described in Proposition 4.7. (For instance, p > [L : Q]N 2 suffices.) In particular, by restricting one of the homomorphisms Πq (x) to g(1) (OL,q ) = q · g(OL,q ) and applying the orbit method map Ω, we get an irreducible character Ξq (x) = Ω(Πq (x)|g(1) (OL,q ) ) of the 1st principal congruence subgroup G(1) (OL,q). When the prime q is clear from the context, it may be dropped from the notation. For every finite extension K ⊂ L and every q ∈ Spec(OL ), the set X (Lq ) is an open subset of Lqdim G+1 . We normalize the additive Haar measure on Lq so that the ring of integers OL,q has measure 1, and denote by λ the product measure on Lqdim G+1 . In [24, Lemma 4.1 and Corollary 4.6], Jaikin–Zapirain proved the following result. Theorem 5.4. There exist quantifier-free definable functions ϕ1 , ϕ2 : X → Γ such that, for every finite extension K ⊂ L, almost all q ∈ Spec(OL ), and every x ∈ X (Lq ), ϕ1 (x) (1) λ(Π−1 , q (Πq (x))) = |OL /q| ∗ (1) (2) dim Ξq (x) = |Ad (G (OL,q))(Πq (x))|1/2 = |OL /q|ϕ2(x) . Remark 5.5. More explicitly, one can take ϕ1 (x) = dim G val(zx ), and ϕ2 (x) = 1 val(α(x)), where α is essentially the function appearing in [24, Corollary 4.6]. According 2 to its definition in [24], the function α, and hence ϕ2 , is quantifier-free. We need a generalization of the construction leading to Theorem 5.4. Employing the notation introduced in Sections 4.3 and 4.4 (compare, in particular, Definition 4.11), the 42 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL definable set GrLie (gk ) is the Grassmannian of Lie subalgebras of gk , and the definable set Grnilp Lie (gk ) is the subset of GrLie (gk ) parametrizing Lie subalgebras that consist of nilpotent matrices. By applying Proposition 4.12, part (1), we see that both sets are, in fact, quantifier-free definable with respect to the theory T fields,OK . e Suppose that R : X → Grnilp Lie (gk ) is a definable function. We denote by R ⊂ X × gO the definable set of tuples (x, X) such that the reduction of X to gk is in R(x). Recall that G ⊂ GLN . By Observation 5.3, we may assume that the residue field characteristic p satisfies p > N so that, over the residue field, one can evaluate without problems the logarithm series up to its Nth term. We define exp R ⊂ X × Gk to be the definable set of pairs (x, g) such that g is unipotent and log g ∈ R(x). By Observation 5.3, we may assume that the residue field characteristic p is unramified in L and satisfies p − 1 > N. In this case, a result of Lazard implies that log(g) converges for elements g of any pro-p e x the fiber of R e at x, we define subgroup of G(OL,q), cf. [26, Lemma B.1]. Denoting by R e ⊂ X × GO to consist of all pairs (x, g) such that the reduction of g to Gk is exp R e x . More generally, if S ⊂ X × Gk is a definable family over X , unipotent and log(g) ∈ R let e S ⊂ X × GO be the definable set of all pairs (x, g) such that the reduction of g to Gk e = exp ^ lies in Sx . We observe that exp R R and will use the lighter notation. By Observation 5.3, for every finite extension K ⊂ L, for almost all primes q of OL , e x (Lq ) is closed under Lie commutators and for every x ∈ X (Lq ), the additive group R e x (Lq ). and it is the Lie ring associated to the pro-p group exp R e x (Lq ). For almost all Definition 5.6. Denote by ΠR,q (x) the restriction of Πq (x) to R primes q, the orbit method map applied to ΠR,q (x) yields an irreducible character ΞR,q (x) e x (Lq ). When the prime q is clear from the context, it may be dropped of the group exp R from the notation. Note that, if R : X → Grnilp Lie (gk ) is the constant function with common value {0}, then the irreducible character Ξ{0},q (x) coincides with the previously defined Ξq (x). For general R, we are interested in possible extensions of the character ΞR,q (x) to its stabilizer e x (Lq )). This stabilizer is known as the inertia group in the normalizer NG(OL,q ) (exp R of ΞR,q (x). ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 43 We summarize the described set-up for x ∈ X (Lq ) in the following diagram. characters groups G(OL,q ) • ΞR,q (x) e x (Lq ) exp R •o Ξq (x) G(1) (OL,q ) • exp-log / correspondence Lie lattices functionals • g(OL,q ) Πq (x) • e x (Lq ) R ΠR,q (x) • g(1) (OL,q ) Πq (x)|g(1) (OL,q ) 5.2. The Stabilizer of ΞR . Suppose now that R : X → Grnilp Lie (gk ) is a quantifier-free definable function with respect to the theory T Hen,K,0. This means that R can be described by a quantifier-free definable formula for all models Lq with sufficiently large residue field characteristic. We remark that many of the results in this section do not yet depend essentially on R being quantifier-free, but it is convenient to focus on this situation in preparation of Section 5.3, where the extra condition plays a crucial role. By Proposition 4.12, parts (3) and (5), there are quantifier-free definable sets NR,Lie ⊂ X × gk and NR ⊂ X × Gk whose fibers over a point x ∈ X (Lq ) are the stabilizers of R(x) under the adjoint actions in the Lie algebra and in the group respectively. By Proposition 4.7, part (3), the stabilizer of ΞR,q (x) in G(OL,q ) is equal to the stabilizer e x (Lq )-orbit of ΠR,q (x). Hence, this stabilizer is the product of in G(OL,q ) of the exp R e x (Lq ) and the stabilizer of ΠR,q (x) in G(OL,q). Writing exp R we have e x (Lq )) = {g ∈ G(OL,q ) | Ad(g)(R e x (Lq )) = R e x (Lq )}, NG(OL,q ) (R StabG(OL,q ) (ΠR,q (x)) = n o e x (Lq )) | ∀Y ∈ R e x (Lq ) : ΠR,q (x)(Y ) = ΠR,q (x)(Ad(g)Y ) . g ∈ NG(OL,q ) (R In the following we consider a prime q of OL , lying above a prime p of OK and different from the previously-omitted primes; see Observation 5.3. In addition we fix an element x = (Ax , zx ) ∈ X (Lq ). Recall that redO|k : O → k denotes the reduction map (applied component-wise to entries of a matrix or vector). Let b S=b SR,q,x ⊂ GO be the definable group, in T Hen,K,0, given by the formula 
 e x val (hAg − Ax , Zi) > val(zx ) , (5.1) ϕ(x, g) =def redO|k (g) ∈ (NR )x ∧ ∀Z ∈ R x where we think of x as a parameter and g as a “free variable”. 44 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL We denote by S = SR,q,x the reduction of b S modulo the maximal ideal, i.e., the definable subgroup of Gk , in T Hen,K,0, given by the formula
(5.2) ψ(x, h) =def ∃g ∈ GO redO|k (g) = h ∧ ϕ(x, g) , where again we think of x as a parameter and h as a “free variable”. Denote the characteristic of the residue field OK /p by p. The functor of p-Witt vectors F 7→ Witt(F) = lim Wittn (F) ←− associates to every perfect field F of characteristic p canonically a strict p-ring Witt(F) with residue field F; see [40, Chapter II]. The integral domain Witt(F) is complete and Hausdorff with respect to the p-adic topology and Wittn (F) ∼ = Witt(F)/pn Witt(F) is called the ring of truncated Witt vectors of length n. For short we denote by FWitt(F) the field of fractions of Witt(F). Q 1 The functor Witt is pro-representable; the underlying set is represented by ∞ i=1 A ; b=S b R,q,x cf. [18]. Using this fact, one sees that there are a pro-algebraic group scheme S and a definable group S = SR,q,x , in T perf.-fields,p,OL/q , such that, for every (possibly infinite) perfect extension F of OL /q, b S(F) =b S(FWitt(F)) ⊂ GO (FWitt(F)), S(F) = S(FWitt(F)) ⊂ Gk (FWitt(F)) ∼ = G(F). (5.3) Our next goal, Proposition 5.7, is to show that there exists an algebraic group S over OL /q such that, for every perfect extension F of OL /q, one has S(F) = S(F). The following table summarizes some of the notation that will feature in this discussion. T Hen,K,0-def. b S ⊂ GO S ⊂ Gk T perf.-fields,p,OL/q -def. S Wittn (F)-algebraic F-(pro-)algebraic Sn b = lim Fn (Sn ) S ←− S We briefly recall further details regarding the functor Witt; compare [18], or [5, p. 276] for a summary. As above, let F be a perfect field of characteristic p > 0 and fix n ∈ N. For an F-scheme X, let Gn (X) be the locally ringed space whose underlying topological space is the same as the topological space of X and whose sheaf of rings is the sheaf of germs of morphisms X → Wittn . For example, Gn (Spec(F)) = Spec(Wittn (F)) and Gn (Spec(F[e]/(e2 ))) = Spec(Wittn (F)[e]/(e2 )), which we will shortly put to good use. By the main theorem of [18, §4], there is a functor Fn : (Wittn (F)-schemes of finite type) → (F-schemes of finite type), ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 45 the Greenberg functor of degree n, such that, for every F-scheme X of finite type and every Wittn (F)-scheme Y of finite type, there is a natural bijection (5.4) HomSpec(F) (X, Fn (Y)) ∼ = HomSpec(Witt (F)) (Gn (X), Y). n Returning to the situation at hand, by Observation 5.3, we may suppose that pZ = q∩Z is unramified in Q ⊂ L. For n ∈ N, there is an affine group scheme Sn over Wittn (OL /q) such that, for every perfect extension F of OL /q,  Sn (Wittn (F)) = g ∈ G(Wittn (F)) | redWittn (F)|F (g) ∈ (NR )x (F) ∧
e x (FWitt(F)) val(hAg − Ax , Zi) > min{n, val(zx )} . ∀Z ∈ R x Here redWittn (F)|F : Wittn (F) → Witt1 (F) ∼ = F denotes the natural reduction map. The b b pro-algebraic group scheme S = SR,q,x in (5.3) is the inverse limit of the OL /q-group schemes Fn (Sn ), n ∈ N. b of S. b We use the following consequence Next we discuss the pro-Lie algebra T of (5.4). If V = Spec(Wittn (F)[x1 , . . . , xd ]/(f1 , . . . , fm )) is an affine Wittn (F)-scheme and v : Spec(Wittn (F)) → V a Wittn (F)-point, then the tangent space of Fn (V) at Fn (v) is the affine subspace of Fn (AdWittn (F) ) ∼ = And F defined by the “polynomials” Fn (dfi (v)), i ∈ {1, . . . , m}. This can be seen from applying (5.4) to X = Spec(F[e]/(e2 )) and Y = V. In particular, the Lie algebra of Fn (Sn ) is isomorphic to Fn (Tn ), where Tn denotes the Wittn (OL /q)-scheme satisfying, for every perfect extension F of OL /q,  Tn (Wittn (F)) = X ∈ g(Wittn (F)) | redWittn (F)|F (X) ∈ (NR,Lie )x (F) ∧
e x (FWitt(F)) val(h[Ax , X], Zi) > min{n, val(zx )} . ∀Z ∈ R b is the inverse limit of the OL /q-Lie algebra schemes Fn (Tn ), n ∈ N. The pro-Lie algebra T Next we show that the definable group S in (5.3) is actually an algebraic group. Proposition 5.7. Let q ∈ Spec(OL ), x ∈ X (Lq ) and S = SR,q,x be as above. In particular, suppose that q satisfies the conditions listed in Observation 5.3. Then S is (equivalent to) an algebraic group over OL /q. Proof. Recall that G ⊂ GLN . We show that S is quantifier-free in T perf.-fields,p,OL/q , using Remark 4.14. Since constructable subgroups are Zariski-closed, this implies that S is (equivalent to) a Zariski-closed subgroup of the algebraic group GLN over OL /q. We need to check that (5.5) S(F) = S(Falg ) ∩ G(F) for every (possibly infinite) perfect extension F of OL /q. Fix such an extension F, set O = Witt(F) with residue field O/pO ∼ = F, and write L = FWitt(F). Let Ounr = Witt(Falg ) unr alg and L = FWitt(F ) denote the maximal unramified extensions. Since Lq is unramified over Qp , we have Lq ⊂ L. 46 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL The inclusion ⊂ in (5.5) is clear. To prove the other inclusion we consider A ∈ g(O), g ∈ G(F), and z ∈ O r {0}. Writing x = (A, z) ∈ X (L) and γ = val(z), we suppose that there exists e g ∈ G(Ounr ) such that redO|k (e g ) = g and val(hAeg − A, Xi) ≥ γ + 1 for all e x (Lunr ). The task is to produce g ∈ G(O) with the same properties as e X∈R g . Clearly, g1 there exists g1 ∈ G(O) such that redO|k (g1 ) = g. Put B = A ∈ g(O) and consider the definable set n o (1) g e Y = YA,B,R(x),γ = g ∈ GO | ∀Z ∈ Rx (val(hB − A, Zi) ≥ γ + 1) , unr e x . Clearly, g −1e where the labeling is permissible, because R(x) determines R ), 1 g ∈ Y(O and it suffices to show that Y(O) is not empty. Furthermore, by forming the quotient of Y (γ+1) by the (γ + 1)st principal congruence subgroup GO of GO , we obtain a Wittγ+1 (OL /q)scheme. Using the Greenberg functor, this quotient can be identified with an algebraic variety Y = YA,B,R(x),γ over OL /q. Our aim Y(O) 6= ∅ is equivalent to Y(F) 6= ∅. e x gives It is convenient to treat first the special case R(x) = {0}. This means that R the 1st principal congruence Lie sublattice g(1) . Since the form h·, ·i is non-degenerate, the defining condition of Y is equivalent to B g ≡ A (mod pγ ). By induction on γ, we may further assume that B ≡ A (mod pγ−1 ), that is B = A + pγ−1 D for some D ∈ g(O). By Observation 5.3, the logarithm map is a well-defined polynomial map on G(1) (Ounr ). For g ∈ G(1) (Ounr ), the formula (5.6) ∞ X 1 B =B+ [B, log(g), . . . , log(g)] ≡ B + [B, log(g)] (mod pδ+1 ), | {z } i! i=1 g i where δ = sup{d ∈ N | [B, log(g)] ≡ 0 (mod pd )}, shows that the defining condition of Y can be replaced by [A, log(g)] + pγ−1 D ≡ 0 (mod pγ ). Passing to Y, this translates into a system of linear equations over F. Since Y(Falg ) is non-empty we deduce that Y(F) is non-empty. Now we return to the general case. Based on the trivial inclusion {0} ⊂ R(x), the special case yields g2 ∈ G(1) (O) such that B g2 ≡ A (mod pγ ). Thus we may assume that B itself is already of the form B = A + pγ E for some E ∈ g(O). By choosing a basis e x (L), the defining condition of Y can be phrased as ℓ(B g − A) ≡ 0 for the Lie lattice R (mod pγ+1 ) for a linear operator ℓ = ℓR(x) : g → g. The elementary divisors of ℓ are 1 and p, with multiplicities dim R(x) and dim gk − dim R(x). Thus (5.6) implies that the defining condition of Y can be replaced by ℓ([A, log(g)] + pγ E) ≡ 0 (mod pγ+1 ), because necessarily [A, log(g)] ≡ 0 (mod pγ ) and all higher terms vanish modulo pγ+1 . Passing to Y, this translates once more into a system of linear equations over F. Since Y(Falg ) is non-empty we deduce that Y(F) is non-empty.  ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 47 Remark 5.8. The proof of Proposition 5.7 admits the following short interpretation. In the special case R(0) = {0}, one can regard Y = YA,B,R(x),γ as a torsor of the connected unipotent group n o (γ+1) (1) (γ+1) UA,γ = gGO ∈ GO /GO | Ag − A ≡ 0 (mod pγ ) , and the logarithm map sets up a bijection between Y and the affine space (γ+1) {Z + gO (1) (γ+1) ∈ gO /gO | [A, Z] ≡ 0 (mod pγ )}. It is known that connected unipotent groups have trivial first Galois cohomology groups, and thus every torsor over such a group has a rational point. We denote the algebraic group equivalent to S = SR,q,x by S = SR,q,x and refer to it as the stabilizer of ΞR,q (x) modulo the 1st principal congruence subgroup. Proposition 5.9. There is a constant C ∈ R, depending only on K and G, such that, for every finite extension K ⊂ L, almost all primes q of OL , every quantifier-free definable function R : X → Grnilp Lie (gk ), and every x = (Ax , zx ) ∈ X (Lq ), the number of connected components of S = SR,q,x is less than C. Proof. Recall that the notation ac×n , respectively val×n , indicates that the angular component map, respectively valuation map, is to be applied coordinatewise to vectors of length n. We apply partial elimination of quantifiers in the theory T Hen,K,0 (cf. Theorem 4.18) by treating the entries of x, which involve elements of Lq , as parameters: after omitting finitely many primes, the formula (5.2) defining S = SR,q,x is equivalent to a formula of the form η(x, h) =def M _ i=1
Hi (Ax ) = 0 ∧ ϕi (ac×m(i) (Hi′ (Ax )), R(x), h) ∧ ψi (val×n(i) (Hi′′ (Ax )), val(zx )) , ′ ′ ′′ ′′ , . . . , Hi,m(i) ) and Hi′′ = (Hi,1 , . . . , Hi,n(i) ) are polynomial funcwhere the Hi , Hi′ = (Hi,1 tions over K (of sort F), the ϕi are formulae in the language of rings (of sort k), and the ψi are formulae in the language of ordered groups (of sort Γ). This can be proved by induction on the length of the formula. that S is a finite union of some of the definable sets  It follows h | ϕi (ac×m(i) (Hi′ (Ax )), R(x), h) . Write S = SR,q,x and Fq = OL /q. Since S(Fq ) always contains the identity, it is non-empty. Proposition 4.15 implies that there is a constant C ∈ R such that, for every unramified finite extension Lq ⊂ Mr with residue field Fqr , there exists d ∈ N0 such that C −1 q rd ≤ |S(Mr )| = |S(Fqr )| ≤ Cq rd . Using Proposition 5.7, we regard S as an algebraic variety S over Fq and apply the Lang– Weil bound [27]. There are infinitely many r ∈ N such that all absolutely irreducible components of S are defined over Fqr , and |S(Fqr )| = (|S : S◦ | + O(q −r/2 ))q r dim S for 48 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL such r. Comparing the two estimates for |S(Fqr )| = |S(Fqr )| as r tends to infinity, we obtain |S : S◦ | ≤ C.  Next we identify the Lie algebra of S = SR,q,x , where R and x = (Ax , zx ) ∈ X (Lq ) are b=T bR,q,x ⊂ gO be the definable set given by as above. In analogy to (5.1), let T   e ξ(x, X) =def redO|k (X) ∈ (NR,Lie )x ∧ ∀Z ∈ Rx (val(h[Ax , X], Zi) > val(zx )) , where we think of x as a parameter and X as a free variable. Furthermore, let T = b modulo the maximal ideal, a definable subset of gk ; TR,q,x denote the reduction of T compare (5.2). Arguing similarly as in the proof of Proposition 5.7, we see that for every finite extension K ⊂ L, almost all primes q of OL , and every x ∈ X (Lq ), the set T(Lq ) is a linear space over the residue field of Lq . In fact, we give an independent proof of this fact in Corollary 5.15. Proposition 5.10. For every finite extension K ⊂ L, almost all primes q of OL , and every x ∈ X (Lq ), the definable set TR,q,x is (equivalent to) the Lie algebra of SR,q,x . Proof. Write x = (Ax , zx ) and γ = val(zx ). We drop all subscripts R, q, x. Recall the b = lim Fn (Sn ) and its pro-Lie algebra T b = construction of the pro-algebraic group S ←− b →S lim Fn (Tn ) via Witt vectors. The reduction map redO|k : b S → S translates into S ←− which factors through the homomorphism fγ : Fγ (Sγ ) → S of OL /q-algebraic groups. By definition, this homomorphism is onto and therefore flat; see [14, Proposition 6.1.5]. As indicated in Remark 5.8, the fibers of fγ , each isomorphic to the kernel of fγ , are affine spaces, and hence smooth. By [15, Theorem 17.5.1], the map fγ is smooth, and so its differential at 1 is surjective. It follows that Fγ (Tγ ) maps onto the Lie algebra T of S. In this way we can identify T with T.  Using Proposition 5.9 and Lemma 4.16, we obtain the following consequence. Corollary 5.11. There is a constant C ∈ R such that, for every finite extension K ⊂ L, almost all primes q of OL , and every x ∈ X (Lq ), C −1 |TR,q,x (Lq )| ≤ |SR,q,x (Lq )| ≤ C|TR,q,x (Lq )|. 5.3. The Lie Algebra Associated to the Stabilizer of ΞR . We continue to work in the set-up introduced in Sections 5.1 and 5.2. In particular, we consistently omit finitely many primes, as specified by Observation 5.3 and in the proof of Proposition 5.9. It is easier, and more transparent, to handle the Lie algebra associated to the stabilizer SR,q,x of ΞR,q (x) modulo the 1st principal congruence subgroup, rather than the stabilizer itself. For this purpose, we introduce the following cover of X . Definition 5.12. Let Y ⊂ X ×(Γ∪{∞})dim g ×(Aut(g)O )2 be the quantifier-free definable set consisting of tuples ((A, z), (γ1 , . . . , γdim g ), (U1 , U2 )) such that, in the chosen standard basis of g, the linear operator T = U1 (ad A) U2 is diagonal and the valuations of the diagonal elements are given by (γ1 , . . . , γdim g ). ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 49 Let R : Y → Grnilp Lie (gk ) be a definable function. The following definitions are analogous e ⊂ Y × gO be the definable set of tuples (y, X) such to those in Section 5.1. Let R that the reduction of X to gk is in R(y). Recall that G ⊂ GLN and that, by virtue of Observation 5.3, the residue field characteristic p satisfies p > N. We define exp R ⊂ Y × Gk to be the set of pairs (y, g) such that g is unipotent and log g ∈ R(y). Finally, e ⊂ Y × GO to consist of all pairs (y, g) again based on Observation 5.3, we define exp R ey. such that the reduction of g to Gk is unipotent and log g ∈ R We denote by prY ↓X : Y → X the natural projection. For every finite extension e y (Lq ) K ⊂ L, almost all primes q of OL , and for every y ∈ Y (Lq ), the additive group R e y (Lq ). is closed under Lie commutators and it is the Lie ring of the pro-p group exp R e y (Lq ) and we Given y ∈ Y (Lq ), we write ΠR,q (y) for the restriction of Πq (prY ↓X (y)) to R e y (Lq ). As in the analogous denote by ΞR,q (y) the resulting irreducible character of exp R Definition 5.6, the subscript q is sometimes omitted. Similarly, we write SR,q,y = SR,q,x , where x = prY ↓X (y). The following lemma is evident. Lemma 5.13. Let O be a complete, discrete valuation ring with a uniformizer ̟. Let M = On be a free O-module of rank n ∈ N, and let N ⊂ M/̟M be a linear subspace with pre-image N in M. Assume that T is an endomorphism of M which, in the standard basis, is given by a diagonal matrix with diagonal entries ̟ γ1 , . . . , ̟ γn , where γ1 , . . . , γn ∈ Z satisfy 0 ≤ γ1 ≤ . . . ≤ γn . For l ∈ Z, let i(l) = max({0} ∪ {i | γi ≤ l}) and j(l) = min({n + 1} ∪ {j | γj ≥ l}). Then the following hold. (1) The pre-image T −1 (̟ l N) is equal to n (a1 , . . . , an ) ∈ M | ∀i ∈ {1, . . . , i(l)} : val(ai ) ≥ l − γi o γi(l) −l γ −l 1 and (̟ a1 , . . . , ̟ ai(l) , 0, . . . , 0) ∈ N . (2) The reduction of T −1 (̟ l N) modulo ̟ is the set of all (a1 , . . . , an ) ∈ (O/̟O)n such that a1 = . . . = aj(l)−1 = 0 and (0, . . . , 0, aj(l), . . . , ai(l) , 0, . . . , 0) ∈ N . Proposition 5.14. Let R : Y → Grnilp Lie (gk ) be a quantifier-free definable function. Then there is a quantifier-free definable function L : Y → GrLie (gk ) such that, for every finite extension K ⊂ L, almost all primes q of OL , and every y ∈ Y (Lq ), the Lie algebra of the stabilizer SR,q,y is given by L(y). Proof. For every y ∈ Y (Lq ), the Lie algebra of the stabilizer SR,q,y is the sum of R(y) and the Lie algebra of the stabilizer of ΠR,q (y) modulo the maximal ideal. Thus it suffices to prove that there is a quantifier-free definable function L : Y → GrLie (gk ) such that the image L(y) of y ∈ Y (Lq ) is the Lie algebra of the stabilizer of ΠR,q (y) modulo the maximal ideal. For y = ((Ay , zy ), (γy,1, . . . , γy,dim g ), (Uy,1 , Uy,2 )) ∈ Y (Lq ), this is the intersection of the normalizer of R(y), which is given by a quantifier-free function, and the reduction 50 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL modulo the maximal ideal of n
o e y (Lq ) ΠR,q (y)([X, Y ]) = 1 Vy (Lq ) = Y ∈ gO (Lq ) | ∀X ∈ (R) n
o e = Y ∈ gO (Lq ) | ∀X ∈ (R)y (Lq ) val(hAy , [X, Y ]i) ≥ val(zy ) n
o e y (Lq ) val(hX, [Ay , Y ]i) ≥ val(zy ) . = Y ∈ gO (Lq ) | ∀X ∈ (R) The set Vy can be interpreted as the fiber of a definable set V ⊂ Y × gO , and it remains to prove that the reduction of Vy is quantifier-free. Let R⊥ (y) be the orthogonal subspace to R(y) in gk with respect to the form h·, ·i. Recall that by omitting finitely many q, we arranged that h·, ·i is non-degenerate on gk (Lq ). We observe that Vy (Lq ) is the pre-image f⊥ )y (Lq )
under the map ad(Ay ). According to the definition of Y , we have of zy (R −1 −1 ad(Ay ) = Uy,1 Ty Uy,2 , where Uy,1 , Uy,2 ∈ Aut(g)O (Lq ) and Ty is diagonal with respect to f⊥ )y (Lq )
under Ty . the standard basis. Thus U −1 (V (Lq )) is the pre-image of zy Uy,1 (R y,2 By Lemma 5.13, statement (2), its reduction modulo the maximal ideal is quantifier-free. Hence, the reduction of Vy is quantifier-free.  Corollary 5.15. For every quantifier-free definable function R : X → Grnilp Lie (gk ), there is a quantifier-free definable function L : X → GrLie (gk ) such that for every finite extension K ⊂ L, almost all primes q of OL , and every x ∈ X (Lq ), the Lie algebra of the stabilizer SR,q,x is given by L(x). Proof. Pre-composing R with the projection prY ↓X : Y → X we get a quantifier-free definable function R′ : Y → Grnilp Lie (gk ). Applying Proposition 5.14, we obtain a quantifier′ free definable function L : Y → GrLie (gk ) such that, for all y ∈ Y (Lq ), the vector space L′ (y) is the Lie algebra of the stabilizer SR,q,y . Since L′ (y) depends only on prY ↓X (y), we get a definable function L : X → GrLie (gk ) such that L(x) is the Lie algebra of the stabilizer SR,q,x for all x ∈ X (Lq ). It suffices to show that L is quantifier-free definable. By an analogue of Lemma 4.13 for valued fields (cf. Remark 4.14), it is enough to show that if F ⊂ E is an extension of valued fields, x ∈ X (F ), and v ∈ GrLie (gk )(F ), then L(x) = v holds in F if and only if L(x) = v holds in E. Since the map Y → X is onto, we can choose y ∈ Y (F ) ⊂ Y (E) such that prY ↓X (y) = x. Then the following assertions are pairwise equivalent: L(x) = v holds in F ; L′ (y) = v holds in F ; L′ (y) = v holds in E (because L′ is quantifier-free); L(x) = v holds in E.  Proposition 5.16. Let R1 , R2 : X → Grnilp Lie (gk ) be quantifier-free definable maps and assume that, for every x ∈ X , the Lie algebra R1 (x) is normalized by R2 (x). Then there is a quantifier-free definable function ϕ : X → Γ such that, for every finite extension K ⊂ L, almost all primes q of L, and every x ∈ X (Lq ), f2 (x))(ΠR ,q (x)) = |OL /q|ϕ(x) . Ad∗ (exp R 1 ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 51 Proof. Similarly to the proof of Proposition 5.14, the first claim of Lemma 5.13 gives a quantifier-free function ϕ1 : Y → Γ such that, for every y ∈ Y (Lq ), f2 (pr(y)))(ΠR pr,q (y)) = |OL /q|ϕ1 (y) , Ad∗ (exp R 1 where pr = prY ↓X : Y → X is the projection. Since ϕ1 (y) depends only on the image of y in X , we get a definable function ϕ2 : X → Γ such that f2 (x))(ΠR ,q (x)) = |OL /q|ϕ2 (x) . Ad∗ (exp R 1 An argument similar to the one in Corollary 5.15 shows that ϕ2 is a quantifier-free definable function, so we can take ϕ = ϕ2 .  Remark 5.17. We can now indicate a proof of part (1) of Theorem 5.4. We write x = (Ax , zx ). Then Π−1 {0} (Π{0} (x)) consists of the pairs (B, w) such that, for 1 all X ∈ g (OL,q),   Ax B − ,X > 0, val zx w or, equivalently, such that val(Ax w − Bzx ) ≥ val(zx w). The claim follows. 5.4. Proof of Theorem 5.1. We continue to work in the same set-up as in the previous sections and recall that, if S ⊂ X × Gk is a definable family over X , then e S ⊂ X × GO denotes the definable set of all pairs (x, g) such that the reduction of g to Gk lies in Sx . Theorem 5.18. There are, for n ∈ N0 , quantifier-free definable functions gn , hn : X → Γ, Rn : X → Grnilp Lie (gk ), Ln : X → GrLie (gk ), constants Cn ∈ R, and definable families Sn ⊂ X × Gk of subgroups of Gk such that the following hold. (1) R0 is the constant function {0}, L0 is the constant function gk , and S0 = X × Gk . (2) For n ≥ 1, every finite extension K ⊂ L, almost all primes q of OL , and every x ∈ X (Lq ), the following hold: fn )x is the stabilizer of ΞR (x) in (Sg ◦ (S n−1 )x , n−1 ◦ Ln (x) is the Lie algebra of (Sn )x , and ◦ Rn (x) is the Lie subalgebra of nilpotent matrices in the solvable radical of Ln (x). (3) There exists n0 ∈ N such that the sequences gn , hn , Rn , Ln , Sn stabilize for n ≥ n0 . (4) For every n, every finite extension K ⊂ L , and almost all primes q of OL , Z |OL /q|gn(x)−hn (x)s ζ(Sfn )x (Lq )|ΞR (x) (s) dλ(x). ζG(OL,q ) (s) − ζG(OL /q) (s) ∼Cn X (Lq ) n Proof. We first construct Rn , Ln , Sn using recursion on n ∈ N0 . Suitable functions gn , hn will be obtained in a second step. The functions R0 , L0 and the family S0 are prescribed by (1). Suppose Rn , Ln , Sn have been constructed. The discussion at the beginning of Section 5.2 implies that there is a definable family of subgroups of GO over X whose 52 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL fn )x . Take Sn+1 to be the reduction of fiber at any x ∈ X is the stabilizer of ΞRn (x) in (S this family modulo the maximal ideal. Similarly, we get Ln+1 , using Corollary 5.15, and Rn+1 , using Proposition 4.12. Thus (2) is taken care of. Next, we show that the sequences Rn , Ln , and Sn , n ∈ N0 , stabilize as required by (3). Note that the sequence dim Rn , n ∈ N0 , is (pointwise) non-decreasing and the sequence dim Ln , n ∈ N0 , is non-increasing. Proposition 5.9 implies that, for every n ∈ N0 , there is an upper bound D(n) for the number of connected components of each of the groups (Sn )x . We claim that, if n1 ∈ N0 is such that dim Rn1 (x) = dim Rn1 +D(n1 ) (x) and dim Ln1 (x) = dim Ln1 +D(n1 ) (x), then the sequences Rn (x), Ln (x), and Sn (x), n ∈ N0 , stabilize for n > n1 + D(n1 ). Indeed, suppose that n ∈ N0 with n1 ≤ n < n1 + D(n1 ). If dim Ln (x) = dim Ln+1 (x), then Ln (x) = Ln+1 (x) and similarly for Rn (x). Since (Sn+1 )x is a subgroup of (Sn )x and they have the same Lie algebra, either Sn (x) = Sn+1 (x) or Sn+1 (x) has fewer connected components than Sn (x). It follows that there is n ∈ N0 with n1 ≤ n < n1 + D(n1 ) such that Sn (x) = Sn+1 (x). It now follows that the sequences of functions Rn , Ln , and Sn stabilize for sufficiently large n ∈ N0 . Once Rn and Sn stabilize, we can keep also the functions gn and hn unchanged. Thus (3) is satisfied. It remains to construct gn and hn for n ∈ N0 so that (4) holds. We start with n = 0. Fix a finite extension K ⊂ L, and consider primes q of OL that satisfy, in particular, the conditions of Lemma 4.6. For short we write X = X (Lq ), and we put G X′ = Xϑ | ϑ ∈ g(1) (OL,q )∨ r {1} , where Xϑ = {x ∈ X | Π{0},q (x) = ϑ} for every homomorphism ϑ from the additive group of the 1st principal congruence Lie sublattice g(1) (OL,q) to C× . Summing over ϑ ∈ g(1) (OL,q)∨ r {1}, applying the orbit method map Ω, and using Lemma 4.8, we obtain ζG(OL,q ) (s) − ζG(OL /q) (s) X 1 = (dim Ω(ϑ))−s ζG(OL,q )|Ω(ϑ) (s) ∗ |Ad (G(OL,q))(ϑ)| ϑ6=1 XZ 1 1 dim(Ξ{0} (x))−s ζG(OL,q )|Ξ{0} (x) (s) dλ(x) = ∗ λ(Xϑ ) |Ad (G(OL,q ))(Π{0} (x))| ϑ6=1 Xϑ Z 1 1 dim(Ξ{0} (x))−s ζG(OL,q )|Ξ{0} (x) (s) dλ(x). = ∗ −1 |Ad (G(O ))(Π (x))| ′ λ(Π (Π (x))) L,q {0} {0} X {0} By Theorem 5.4 and Corollary 5.15, there are quantifier-free definable functions ϕ1 , ϕ2 , ϕ3 : X → Γ such that, for almost all primes q of OL , ϕ1 (x) (1) λ(Π−1 , {0} (Π{0} (x))) = |OL /q| (2) dim Ξ{0} (x) = |Ad∗ (G(1) (OL,q ))(Π{0} (x))|1/2 = |OL /q|ϕ2 (x) , (3) dim L1 (x) = |OL /q|ϕ3 (x) . ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 53 By Lemma 4.16, there is a constant C ∈ R such that, for every x ∈ X, |Ad∗ (G(OL,q ))(Π{0} (x))| = |Ad∗ (G(1) (OL,q))(Π{0} (x))| · |G(OL /q)/(S1)x (Lq )| ∼C |OL /q|2ϕ2 (x)+dim g−ϕ3 (x) . Therefore, ζG(OL,q ) (s) − ζG(OL /q) (s) ∼C Z |OL /q|(−ϕ1 (x)−2ϕ2 (x)+ϕ3 (x)−dim g)−ϕ2 (x)s ζG(OL,q )|Ξ{0} (x) (s) dλ(x), X′ and we set g0 (x) = −ϕ1 (x) − 2ϕ2 (x) + ϕ3 (x) − dim g, h0 (x) = ϕ2 (x), and C0 = C. Finally, we suppose that gn , hn are given for some n ∈ N0 and we explain how to construct gn+1 , hn+1 . Again, fix a finite extension K ⊂ L, and consider primes q of OL that are different from any of the finitely many primes omitted. Let a ⊂ g(OL /q) be a nilpotent Lie algebra, and recall that e a denotes the pre-image of a under the map g(OL,q ) → g(OL /q); it is a Lie ring, and, in particular, an additive group. Let τ be a homomorphism from e a to C× , and consider the set Xa,τ = Π−1 Rn (τ ) = {x ∈ X (Lq ) | Rn (x) = a, ΠRn (x) = τ }. Inductively, we see that on Xa,τ , the values of R0 (x), L0 (x), . . . , Rn (x), Ln (x) and thus the values of (Sn )x (Lq ), (Sn+1 )x (Lq ), and Rn+1 (x) are constant. Denote the latter by Sn , Sn+1, and b respectively. Writing m = |b : a|, let ϑ1 , . . . , ϑm denote the homomorphisms from e b to C× that extend τ . We define, for i ∈ {1, . . . , m}, i Xa,τ = Π−1 Rn+1 (ϑi ) = {x ∈ Xa,τ | ΠRn+1 (x) = ϑi }. These sets form a partition of Xa,τ into m parts of equal measure. Using Lemma 4.8 and Clifford theory, we deduce that, for every x ∈ Xa,τ , ζ(Sfn )x (Lq )|ΞR n (x) −s fn : S ] (s) (s) = ζSfn |Ω(τ ) (s) = |S n+1 | ζS ^ n+1 |Ω(τ ) −s  m ∗ X ] |Ad (exp R )(τ )| dim Ω(ϑ ) n+1 i −s f ] = | Sn : S ζS^ |Ω(ϑ ) (s). n+1 | n+1 i ∗ ] dim Ω(τ ) |Ad ( S )(ϑ )| n+1 i i=1 54 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Therefore, Z Xa,τ ζ(Sfn )x (Lq )|ΞR fn : S ] = |S n+1 | = m Z X i=1 i Xa,τ −s n (x) (s) dλ(x) m Z X i=1 i Xa,τ ] λ(Xa,τ ) |Ad∗ (exp R n+1 )(τ )| ∗ ] λ(X i ) | {za,τ } |Ad (Sn+1 )(ϑi )|  dim Ω(ϑi ) dim Ω(τ ) −s ζS^ (s) dλ(x) n+1 |Ω(ϑ ) i =m |(Sn )x (Lq ) : (Sn+1 )x (Lq )|−s |Rn+1 (x) : Rn (x)| · | {z } ] |Ad (exp R n+1 )(ΠRn (x))| |Ad∗ ((Sg (x))| n+1 )x (Lq ))(ΠR ∗ n+1  =m dim ΞRn+1 (x) dim ΞRn (x) −s ζ(Sg n+1 )x (Lq )|ΞR n+1 (x) (s) dλ(x). Now there are quantifier-free definable functions ψ1 , ψ2 , ψ3 , ψ4 : X → Γ and a constant C ∈ R – all independent of L and q – such that, for almost all primes q of OL , and all x ∈ X (Lq ), the following hold: (1) |(Sn )x (Lq ) : (Sn+1 )x (Lq )| ∼C |OL /q|ψ1 (x) , by Corollary 5.15 and Lemma 4.16 (1), because the number of connected components of (Sn )x is bounded by D(n), (2) |Rn+1(x) : Rn (x)| = |OL /q|ψ2 (x) , as Rn and Rn+1 are quantifier-free definable, ∗ g ] (x))| ∼C |OL /q|ψ3 (x) , as (3) |Ad∗ (exp R n+1 )(ΠR (x))|/|Ad ((Sn+1 )x (Lq ))(ΠR n n+1 |Ad∗ ((Sg n+1 )x (Lq ))(ΠRn+1 (x))| = ] |Ad∗ (exp R n+1 (x))(ΠRn+1 (x))| · |(Sn )x (Lq ) : (Sn+1 )x (Lq )|, by part (1), and by Proposition 5.16, (4) dim ΞRn+1 (x)/ dim ΞRn (x) ∼C |OL /q|ψ4 (x) because, for example, dim ΞRn (x) = fn (x))(ΠR (x))|1/2 and by Proposition 5.16. |Ad∗ (exp R n Writing αn = ψ2 + ψ3 and βn = ψ1 + ψ4 , we obtain Z Xa,τ ζ(Sfn )x (Lq )|ΞR n (x) (s) dλ(x) ∼C 3 Z Xa,τ |OL /q|αn (x)−βn (x)s ζ(Sg n+1 )x (q)|ΞR n+1 (x) (s) dλ(x). ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 55 Defining gn+1 = gn +αn and hn+1 = hn +βn we obtain, from the corresponding properties of gn , hn , Z |OL /q|gn (x)−hn (x)s ζ(Sfn )x (Lq )|ΞR (x) (s) dλ(x) ζG(OL,q ) (s) − ζG(OL /q) (s) ∼Cn n X (Lq ) Z X |OL /q|gn(x)−hn (x)s ζ(Sfn )x (Lq )|ΞR (x) (s) dλ(x) = a,τ ∼C 3 XZ a,τ = Z n Xa,τ Xa,τ X (Lq ) |OL /q|gn+1(x)−hn+1 (x)s ζ(Sg n+1 )x (Lq )|ΞR |OL /q|gn+1(x)−hn+1 (x)s ζ(Sg n+1 )x (Lq )|ΞR n+1 n+1 (x) (s) dλ(x) (x) (s) dλ(x). Here a ranges over the finite set of nilpotent Lie subalgebras of g(OL /q) and τ ranges over the countable set of characters of e a. The required properties of gn+1 , hn+1 hold for 3 Cn+1 = Cn C .  We are now ready to prove Theorem 5.1. Proof of Theorem 5.1. We continue to use the notation set up in this section. In particular, let gn , hn , Rn , Ln , Sn , Cn be the sequences constructed in Theorem 5.18. Suppose n0 ∈ N is sufficiently large so that gn , hn , Rn , Ln , Sn are stable for n ≥ n0 . Fix n ≥ n0 . By Theorem 5.1, for all finite extensions K ⊂ L and almost all q, Z |OL /q|gn (x)−hn (x)s · ζ(Sfn )x (Lq )|ΞR (x) (s)dλ(x). (5.7) ζG(OL,q ) (s) − ζG(OL /q) (s) ∼Cn n X (Lq ) Recall that (Sn )x ⊂ Gk is an algebraic group. Let (Sn )◦x denote the connected compo◦ ] ◦ nent of the identity, and let (S n )x ⊂ GO be the pre-image of (Sn )x under the reduction map modulo the maximal ideal. It was shown in Proposition 5.9 that there is a constant D1 ∈ R such that, for almost all q and all x ∈ X (Lq ), ◦ ] ◦ fn )x (Lq ) : (S |(S n )x (Lq )| ≤ |(Sn )x : (Sn )x | ≤ D1 . By Lemma 2.11 (2), we get for almost all q and all x ∈ X (Lq ), ζ(Sfn )x (Lq )|ΞR n (x) ∼D1 ζ(S ^ )◦ (L n x q )|ΞRn (x) . For every prime q and every x ∈ X (Lq ), the group Qx = (Sn )◦x / exp Rn (x) is a reductive group over the field OL /q of dimension at most dim G. Omitting finitely many primes q of OL as specified in Observation 5.3, none of the Schur multipliers of the ] ◦ ^ groups (S n )x (Lq )/ exp Rn (x)(Lq ) = Qx (OL /q), for x ∈ X (Lq ), contains elements of order char(OL /q). Using Lemmas 4.2 and 4.1, we thus obtain ζ(Sg ◦ n ) (Lq )|ΞR x n (x) = ζQx (OL /q) . 56 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL The Lie algebra of Qx is Ln (x)/Rn (x). By Proposition 4.12, there is a finite, quantifierfree partition of X such that, on each part, the absolute root system Φx of Qx /Z(Qx ) and the dimension of the center of Qx – which can be read off from the Lie algebra of Qx , cf. Proposition 4.12 – are constant. By Corollary 3.4, there is a constant D2 ∈ R such that + ζQx (OL /q) (s) ∼D2 |OL /q|dim Z(Qx ) (1 + |OL /q|rk Φx −|Φx |s ). Setting Z = X × {0, 1}, ( gn (x) + dim Z(Qx ) for z = (x, 0) ∈ Z , f1 (z) = gn (x) + dim Z(Qx ) + rk Φx for z = (x, 1) ∈ Z , and ( −hn (x) f2 (z) = −hn (x) + |Φ+ x| we obtain, with C = Cn D1 D2 , for z = (x, 0) ∈ Z , for z = (x, 1) ∈ Z , ζG(OL,q ) (s) − ζG(OL /q) (s) Z + ∼C |OL /q|gn(x)+dim Z(Qx )+hn (x)s (1 + |OL /q|rk Φx −|Φx |s )dλ(x) X (Lq ) Z = |OL /q|f1(z)−f2 (z)s dλ(z). Z (Lq ) This completes the proof of Theorem 5.1.  6. Quantifier-Free Integrals In this section we complete the proof of Theorem 2.8. The section’s main result is Theorem 6.2, expressing the dependence of an integral such as the one in Theorem 5.1 on the local field the integral is interpreted at. We state the precise result in Section 6.1 and prove it in Section 6.2. Throughout this section, we fix a number field K and work within the first order language of valued fields together with a constant, of the value field sort, for every element of K. We will use the theory T Hen,K,0 of Henselian valued fields over K of characteristic 0; cf. Definition 4.17. 6.1. Uniform Formulae for Quantifier-Free Integrals. We make no notational distinction between an algebraic variety and the corresponding functor of points. Definition 6.1. Let X be a smooth algebraic variety of dimension n over K and let ω be a regular differential n-form on X. For any local field F containing K, the set X(F ) has the structure of a p-adic analytic manifold; cf. [41, Part II, Chapter III]. We define a measure |ω|F on X(F ) as follows: given a compact open set U ⊂ X(F ), an open compact subset W ⊂ F n , and an analytic diffeomorphism f : U → W , we write f ∗ ω = gdx1 ∧ · · · ∧ dxn , ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH for some function g : W → F , and define |ω|F (U) = Z 57 |g(x)|F dλ(x), W where |·|F is the normalized absolute value of F and λ is the Haar measure on F n normalized so that λ(OFn ) = 1. The assignment U 7→ |ω|F (U) extends uniquely to a non-negative (possibly infinite) Radon measure on X(F ), which we also denote by |ω|F . See [1, Section 3.1] for further details. We now state the section’s main theorem. Theorem 6.2. Let K be a number field with ring of integers OK , and let X ⊂ AM be a smooth affine K-variety, and ω a regular differential top form on X. Suppose that Z ⊂ X ∩ OM is a quantifier-free definable set and that f1 , f2 : Z → Γ are quantifierfree definable functions. There exist N ∈ N, quasi-affine OK -schemes Wi and integers αi , βi ∈ N0 and ni ∈ N, for i ∈ {1, . . . , N}, and Aij , Bij ∈ Z, for i ∈ {1, . . . , N} and j ∈ {1, . . . , ni }, such that the following holds: for every finite extension K ⊂ L and almost all primes q of OL , Z ni N X Y |OL /q|Aij −Bij s αi −βi s f1 (z)−f2 (z)s |OL /q| |Wi (OL /q)|· |OL /q| |ω|Lq = (6.1) , 1 − |OL /q|Aij −Bij s Z (Lq ) i=1 j=1 for every s ∈ C for which the integral on the left converges. Theorem 6.2 may possibly be deduced from [19, Proposition 4.5 and Proposition 10.10] in a similar way that [19, Theorem 1.3] is. In the next section we give a direct proof, in terms of resolutions of singularities. 6.2. Proof of Theorem 6.2. In this section we decorate the reduction, angular component and valuation maps with a superscript to indicate the arity of their domain and write, e.g., red×n , ac×n and val×n (cf. Section 4.4). Definition 6.3. A rational polyhedral cone in Qn is the intersection of finitely many open or closed linear half-spaces, i.e. subsets of the form {x ∈ Qn | hx, vi > 0} or {x ∈ Qn | hx, vi ≥ 0}, where v ∈ Qn and h , i denotes the usual inner product on Rn . We will reduce Theorem 6.2 to the following special case: Special Case. There exist an OK -scheme X ⊂ AM OK such that the structure map X → Spec(OK ) is smooth and its non-empty fibers are irreducible of dimension n, an étale map ξ = (ξ 1 , . . . , ξ n ) : X → AnOK , a rational polyhedral cone D ⊂ Qn≥0 , an OK -scheme M ⊂ (Gnm )OK × X, and integers a1 , . . . , an , b1 , . . . , bn such that (1) X = X × Spec(K), i.e. X is an OK -model of X, (2) ω = ξ ∗ (dx1 ∧ · · · ∧ dxn ), where x1 , . . . , xn are coordinates on AnOK , P (3) f1 (z) = ni=1 ai val(ξ i (z)), P (4) f2 (z) = ni=1 bi val(ξ i (z)), 58 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL (5) the quantifier-free definable set Z is defined by the formula
z ∈ X ∩ OM ∧ val×n (ξ(z)) ∈ D ∧ ac×n (ξ(z)), red×M (z) ∈ Mk . 6.2.1. Proof of Theorem 6.2 in the Special Case. Let Z ⊂ X be the image of M under the projection to X. After reordering the coordinates of the map ξ and passing, if necessary, to one of the parts of a finite, quantifier-free partition of M, we can assume that, for some 0 ≤ t ≤ n, the functions ξ 1 , . . . , ξ t vanish on Z×Spec(K) and ξ t+1 , . . . , ξ n are invertible on Z × Spec(K). This implies that, for every finite extension K ⊂ L and almost all primes q of OL , the functions ξ 1 , . . . , ξ t vanish on Z × Spec(OL /q) and ξ t+1 , . . . , ξ n are invertible on Z × Spec(OL /q). We may thus assume, without loss of generality, that D ⊂ Qt>0 × {0}n−t . For each p ∈ X(OL /q), the map ξ induces a measure-preserving diffeomorphism be   

×n −1 ×M −1 (ξ(p)), |dx1 ∧ · · · ∧ dxn |Lq . The tween (p) ∩ X(Lq ), |ω|Lq and red red
−1 n that (p) ∩ Z (Lq ) under ξ is the set of all x = (x1 , . . . , xn ) ∈ OL,q image of red×M satisfy (1) val×n (x) ∈ D ∩ Zn , (2) red×n (x) = ξ(p), and (3) (ac×n (x), p) ∈ M(OL /q), or, equivalently, (1)’ val×n (x) ∈ D ∩ Zn ,   (2)’ (ac(xt+1 ), . . . , ac(xn )) = ξ t+1 (p), . . . , ξ n (p) , (3)’ (ac(x1 ), . . . , ac(xt )) ∈ M(p) (OL /q), where M(p) (OL /q) = n o (y1 , . . . , yt) ∈ ((OL /q)× )t | (y1 , . . . , yt, ξ t+1 (p), . . . , ξ n (p), p) ∈ M(OL /q) . n Setting W(p) ⊂ OL,q to be the set defined by the conjunction of these three conditions, we get Z 1Z (Lq ) (z)|OL /q|f1 (z)−f2 (z)s |ω|Lq ×M −1 (red ) (p) Z Pn i=1 (ai −bi s) val(xi ) |dx ∧ · · · ∧ dx | 1 (x)|O /q| = W(p) L 1 n Lq ×n −1 red (ξ(p)) ( ) X Pn = M(p) (OL /q) · |OL /q| i=1 (ai −bi s)γi . γ∈D∩Zn ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH Therefore, Z 59 |OL /q|f1 (z)−f2 (z)s |ω|Lq Z (Lq ) Z 1Z (Lq ) (z)|OL /q|f1 (z)−f2 (z)s |ω|Lq X(OL,q ) X Z = 1Z (Lq ) (z)|OL /q|f1(z)−f2 (z)s |ω|Lq −1 ×M ) (p) p∈X(OL /q) (red X X Pn = M(p) (OL /q) · |OL /q| i=1 (ai −bi s)γi . = (6.2) p∈X(OL /q) γ∈D∩Zn n−t X, where the map M → (Gm )OK × X is Let W be the fiber product M ×(Gn−t m )OK ×X n−t the projection onto the last n − t + M coordinates, and the map X → (Gm )OK × X is (ξ t+1 , . . . , ξ n ) × IdX . As X |W(OL /q)| = |M(p) (OL /q)|, p∈X(OL /q) (6.2) implies that Z X Pn |OL /q| i=1 (ai −bi s)γi . (6.3) |OL /q|f1 (z)−f2 (z)s |ω|Lq = |W(OL /q)| · Z (Lq ) γ∈D∩Zn The set D ∩ Zn may be decomposed into a finite, disjoint union of cosets of free monoids. We may thus replace D ∩ Zn by such a coset. Sums as the ones in (6.3) over free monoids |OL /q|A−Bs are just finite products of geometric progressions of the form 1−|O A−BS , for suitable L /q| numerical data A, B ∈ Z. Translating the monoid amounts to multiplying the relevant product by a factor of the form |OL /q|α−βs , for suitable α, β ∈ Z. This proves Theorem 6.2 in the Special Case. 6.2.2. Reduction to the Special Case. Let X, ω, Z , f1 , f2 be as in Theorem 6.2. As X is smooth, the integral (6.1) in the theorem is the sum of the respective integrals over the irreducible components of X. Hence, we can assume that X is irreducible. Lemma 6.4. If h : OM → Γ is a quantifier-free definable function, then there exist N ∈ N, a finite quantifier-free definable partition OM = Ω1 ⊔ · · · ⊔ ΩN , and, for each i ∈ {1, . . . , N}, finitely many Pni polynomials Pi1 , . . . , Pini over K and rational numbers ri1 , . . . , rini such that h|Ωi = j=1 rij val ◦Pij . Proof. By assumption, the graph of h is a Boolean combination of quantifier-free formulae in M + 1 variables x1 , . . . , xM , γ, where xi are valued field sort and γ is value group sort. 60 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL The quantifier-free formulae, in turn, are Boolean combinations of formulae of the form X nj val(Pj (x)) = 0, P (x) = 0, Q(ac(R1 (x)), . . . , ac(RN ′ (x))) = 0, and n0 γ + j∈J where x = (x1 , . . . , xM ), J is a finite index set, P , R1 , . . . , RN ′ , and Pj , j ∈ J, are polynomials over K, Q is a polynomial over ac(K), and n0 and nj , j ∈ J, are integers. This implies the claim.  By Lemma 6.4, we can assume that the functions f1 and f2 have the form X X f1 = rk val ◦Fk , f2 = rk′ val ◦Fk , k∈J k∈J where J is a finite indexing set, ∈ Q, and Fk are regular functions on X. By definition, the definable set Z is a disjoint union of sets that are defined using formulas of the form rk , rk′ (6.4) ′ ′ z ∈ X ∩ OM ∧ ϕ(ac×M (H(z))) ∧ ψ(val×M (H(z))) ∧ H ′ (z) = 0, ′ where H, H ′ : X → AM K are regular maps defined over K, ϕ is a quantifier-free formula in the language of fields (in variables of sort k), and ψ is a quantifier-free formula in the language of ordered groups (in variables of sort Γ). Hence, it is enough to prove the theorem assuming that Z is defined by a formula of the form (6.4). If H ′ 6= 0, then the integral in (6.1) is zero. Hence, we can assume that H ′ = 0. The definable set defined by ϕ is equivalent to the disjoint union of finitely many quasi-affine varieties. Partitioning Z according to these varieties, we can assume that ϕ defines a single quasi-affine K-variety V . Since we are only interested in evaluating the integral (6.1) over local fields with large residue field characteristic, we may replace V by one of its OK models, which we denote by V. We apply a similar argument for ψ: after passing to one of the parts of a finite, quantifier-free partition of Z , we can assume that ψ defines ′ ×M ′ a translation of a rational polyhedral cone in QM (e), ≥0 by a vector of the form val M′ where e ∈ K . For every finite extension K ⊂ L and almost all q ∈ Spec(OL ), we have ′ ′ val×M (e) = 0. We may thus assume that ψ defines a rational polyhedral cone C ⊂ QM ≥0 . In conclusion, we can assume that Z is defined by the formula (6.5) ′ ′ z ∈ X ∩ OM ∧ ac×M (H(z)) ∈ V ∧ val×M (H(z)) ∈ C. QM ′ 6.2.3. Resolution of Singularities. We write H = (H1 , . . . , HM ′ ), let P = i=1 Hi · Q F , and consider the divisor D = div(P · ω), i.e. the union of the vanishing loci k∈J k of ω and P . By Hironaka’s theorem on strict resolution of singularities (cf. [1, Definition B.5.1]), applied to the divisor D, there exist m ∈ N and a smooth variety Y ⊂ X ×Pm defined over K such that the projection π : Y → X is birational, is an isomorphism above the complement of D, and the pullback of D under π is a divisor with normal crossings. Denote the dimension of X S by n. By the definition of divisor with normal crossings, there is an open cover Y = i∈I Ui by affine K-varieties Ui , for some finite index set I, and, for each i ∈ I, there is an étale map ξi : Ui → An such that π −1 (supp(D)) ∩ Ui is ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 61 contained in the pre-image under ξi of the coordinate hyperplanes in An . The divisor of π ∗ ω is supported on π −1 (D), and ξi∗ (dx1 ∧ · · · ∧ dxn ) is an invertible top differential ∗ω is regular and its divisor is supported on π −1 (D). form. Hence, the function ξ∗ (dx1π∧···∧dx n) i Therefore, there is a regular function ϑi : Ui → Gm and aij ∈ Z, j ∈ {1, . . . , n}, such that, for y ∈ Ui , n Y π∗ω a (y) = ϑi (y) · (ξi (y))j ij , ∗ ξi (dx1 ∧ · · · ∧ dxn ) j=1 For each i ∈ I, fix an embedding Ui ⊂ AN for some N. For every finite extension K ⊂ L and almost all primes q of OL , the function ϑi is the restriction of a polynomial in N N variables with coefficients in OL,q. In particular, its restriction to OL,q has non-negative valuation. The same is true for 1/ϑi and so, for almost all q, the restriction of val(ϑi ) to N N Ui (Lq ) ∩ OL,q is 0. Consequently, we obtain, for y ∈ Ui (Lq ) ∩ OL,q ,   n X π∗ω val ∗ aij val(ξi (y)j ). (y) = ξi (dx1 ∧ · · · ∧ dxn ) j=1 Similarly there are, for j ∈ {1, . . . , n} and t ∈ {1, . . . , M ′ }, integers bij , cij , ditj and funcN tions ηit : Ui → Gm , such that, for y ∈ Ui (Lq ) ∩ OL,q , f ◦ π(y) = n X bij val(ξi (y)j ), n X cij val(ξi (y)j ), j=1 j=1 ′ g ◦ π(y) = and, for t ∈ {1, . . . , M }, val(Ht ◦ π(y)) = n X ditj val(ξi (y)j ), ac(Ht ◦ π(y)) = ac(ηit (y)) · j=1 n Y ac(ξi (y)j )ditj . j=1 6.2.4. Reduction Modulo q. Fix a total S ordering <Fon the finite index set I. For i ∈ I set S Zi = Ui r j<i Uj ֒→ AN , so that Y = i∈I Ui = i∈I Zi , the latter union being disjoint. m N Further choose OK -models X ⊂ AM OK , Y ⊂ X × POK , and Zi ⊂ AOK for the varieties X, Y , and Zi , for i ∈ I. We also fix OK -models for the maps ηit , for t ∈ {1, . . . , M ′ }, and ξi which – by slight abuse of notation – we continue to denote by these letters. Lemma 6.5. For every finite extension K ⊂ L and almost all primes q of OL , G π −1 (X(OL,q)) = red−1 (Zi (OL /q)). i∈I Proof. We first contend that π −1 (X(OL,q )) = Y(OL,q) for almost all q. One containment follows from the projectivity of π, the other follows from the fact that, for almost all q, the map π is defined over OL,q . Next, we claim that, for almost all q, the sets Zi (OL /q) form a partition of Y(OL /q). Indeed, we know that Zi (C) form a partition of Y(C), so the Zi ((OL /q)alg ), i ∈ I, form a partition of Y((OL /q)alg ) for almost all q, using Robinson’s 62 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL Principle. Since Zi and Yi are quantifier-free, we get that Zi (OL /q) form a partition of Y(OL /q) for these primes q. Altogether this yields, for almost all q, G G G G Y(OL,q) = red−1 (p) = red−1 (p) = red−1 (Zi (OL /q)). i∈I p∈Zi (OL /q) p∈Y(OL /q) i∈I  We deduce from Lemma 6.5 that, for every finite extension K ⊂ L and almost all primes q of OL , Z Z f1 (z)−f2 (z)s 1Z (Lq ) (x)|OL /q|f1 (z)−f2 (z)s |ω|Lq |OL /q| |ω|Lq = X(OL,q ) Z (Lq ) Z = 1Z (Lq ) (π(y))|OL /q|f1(π(y))−f2 (π(y))s |π ∗ ω|Lq π −1 (X(OL,q )) XZ 1π−1 (Z )(Lq ) (y)|OL/q|f1 (π(y))−f2 (π(y))s |π ∗ ω|Lq . = i∈I red−1 q (Zi (OL /q)) ′ Recall from Section 6.2.2 the rational polyhedral cone C ⊂ QM ≥0 and the OK -model V of the quasi-affine K-variety V , featuring in the definition (6.5) of the quantifier-free definable set Z , as well as the various data defined in Section 6.2.3. For each i ∈ I, let Di ⊂ Qn≥0 be the rational polyhedral cone defined by !M ′ n X (γ1 , . . . , γn ) ∈ Di ⇐⇒ ditj γj ∈ C, j=1 let Mi ⊂ (Gnm )OK t=1 × Y be the OK -scheme defined by (x1 , . . . , xn , y) ∈ Mi ⇐⇒ ηit (y) n Y j=1 d xj itj !M ′ ∈ V, t=1 and let Zi be the definable set defined by Then Z   y ∈ Ui ∩ OM +m ∧ val×n (ξi (y)) ∈ Di ∧ ac×n (ξi (y)), red×(M +m) (y) ∈ (Mi )k . |OL /q|f1 (z)−f2 (z)s |ω|Lq = Z (Lq ) XZ i∈I Zi (Lq ) |OL /q| Pn j=1 (aij +bij −cij s) val(ξi (y)j ) |ξi∗(dx1 ∧ · · · ∧ dxn )|Lq , and each summand on the right hand side is of the form covered by the Special Case. This concludes the proof of Theorem 6.2. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 63 6.3. Proof of Theorem 2.8. Let Φ be the absolute root system of G. We show that the assertions of the theorem hold for c(G) = a(Φ) ∪ b(G), where a(Φ) ∈ A+ is the element constructed in Theorem 3.1 and b(G) is obtained as follows. By Theorem 5.1, there are a quantifier-free definable set Z , quantifier-free definable functions f1 , f2 : Z → Γ, and a constant C1 ∈ R such that, for every finite extension K ⊂ L and almost all primes q of OL , Z |OL /q|f1(z)−f2 (z)s dλ(z). ζG(OL,q) (s) − ζG(OL/q) (s) ∼C1 Z (Lq ) Furthermore, Theorem 6.2 gives that, for almost all primes q, (6.6) ζG(OL,q ) (s) − ζG(OL /q) (s) ∼C1 N X |OL /q| αi −βi s |Wi (OL /q)| · i=1 ni Y |OL /q|Aij −Bij s , Aij −Bij s 1 − |O /q| L j=1 where the W1 , . . . , WN are quasi-affine OK -schemes and ni , Aij , Bij , αi , βi are integers specified in Theorem 6.2. We can assume that the generic fiber of each Wi is non-empty and irreducible, and we set n  o Xn i Xni b(G) = αi + dim Wi + Aij , −βi − Bij | 1 ≤ i ≤ N ∈ A+ . j=1 j=1 Let Q ⊂ Spec(OL ) denote the set of all primes q such that (6.6) holds. By the Lang– Weil estimates [27], there is a constant C2 ∈ R such that, for each i ∈ {1, . . . , N} and for almost all q ∈ Q, either Wi (OL /q) = ∅ or |Wi (OL /q)| 1 ≤ ≤ C2 . 2 |OL /q|dim Wi For each q ∈ Q, there exists i ∈ {1, . . . , N} such that Wi (OL /q) 6= ∅, and we define βq = max {Aij /Bij | 1 ≤ i ≤ N, 1 ≤ j ≤ ni , Wi (OL /q) 6= ∅, Bij 6= 0} ∈ Q>0 , n o Xn i Xn i
bq = αi + dim Wi + Aij , −βi − Bij | 1 ≤ i ≤ N, Wi (OL /q) 6= ∅ ∈ A+ . j=1 j=1 We observe that, for each prime q ∈ Q, the abscissa of convergence of ζG(OL,q ) is equal to βq by (6.6) and bq ⊂ b(G). Let ε ∈ R>0 . Since |OL /q| ≥ 2 for all q ∈ Spec(OL ), there is a constant δ(ε) ∈ R>0 such that, for each i ∈ {1, . . . , N}, each q ∈ Q, and all σ ∈ R with σ > βq + ε, Y
δ(ε) < 1 − |OL /q|Aij −Bij σ ≤ 1. j From (6.6) it follows that there is a constant C3 (ε) ∈ R such that, for every q ∈ Q, (6.7) ζG(OL,q ) − ζG(OL /q) ∼C3 (ε) ξbq ,|OL/q| for σ > βq + ε. 64 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL By Theorem 3.1 and Remark 3.2, there is a constant C4 ∈ R such that, for almost all primes q ∈ Spec(OL ), there is aq ⊂ a(Φ) such that (6.8) ζG(OL /q) − 1 ∼C4 ξaq ,|OL /q| . Combining (6.7) and (6.8), we get that, for almost all primes q ∈ Q, the following holds: for every ε ∈ R>0 , ζG(OL,q ) − 1 ∼2 max{C3 (ε),C4 } ξaq ∪bq ,|OL/q| for σ > βq + ε. These primes form a co-finite subset T (L) ⊂ Spec(OL ), and this proves the assertion (1) of the theorem. Assertion (2) of the theorem is derived from the argument above as follows. The set R1 (L) = {q ∈ Spec(OL ) | aq = a(Φ)} is a Chebotarev set by Corollary 3.15. Moreover, the Chebotarev Density Theorem implies that {q ∈ Q | ∀i ∈ {1, . . . , N} : Wi (OL /q) 6= ∅} is a Chebotarev set. Hence R2 (L) = {q ∈ Q | bq = b(G)} is a Chebotarev set. It follows that R(L) = R1 (L) ∩ R2 (L) is a Chebotarev set, in particular of positive analytic density. This concludes the proof of Theorem 2.8. References [1] Aizenbud, A.; Avni, N.; Representation growth and rational singularities of the moduli space of local systems. To appear in Inventiones Math., preprint: http://arxiv.org/abs/1307.0371. [2] Avni, N.; Arithmetic groups have rational representation growth. Ann. of Math. 174 (2011), 1009–1056. [3] Avni, N.; Klopsch, B.; Onn, U.; Voll, C.; Representation zeta functions of compact p-adic analytic groups and arithmetic groups. Duke Math. J. 163 (2013), 111–197. [4] Avni, N.; Klopsch, B.; Onn, U.; Voll, C.; Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2 . To appear in Proc. London Math. Soc., preprint: http://arxiv.org/abs/1410.4533. [5] Bosch, S.; Lütkebohmert, W.; Raynaud, M.; Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 21. Springer-Verlag, Berlin Heidelberg, 1990. [6] Carter, R. W.; Finite groups of Lie type. Conjugacy classes and complex characters. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. [7] Chang, C. C.; Keisler, H. J.; Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990. [8] Chatzidakis, Z.; Théorie des modèles des corps finis et pseudo-finis. Available at: http://www. logique.jussieu.fr/~zoe/index.html. [9] Chatzidakis, Z.; van den Dries, L.; Macintyre, A.; Definable sets over finite fields. J. reine angew. Math. 427 (1992), 107–135. [10] Cluckers, R.; Loeser, F.; Constructable motivic functions and motivic integration. Invent. Math. 173 (2008), 23–121. [11] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A.; Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham, 1985. [12] Deriziotis, D. I.; Centralizers of semisimple elements in a Chevalley group. Comm. Algebra 9 (1981), 1997–2014. ARITHMETIC GROUPS, BASE CHANGE, AND REPRESENTATION GROWTH 65 [13] Digne, F.; Michel, J.; Representations of finite groups of Lie type. London Mathematical Society Student Texts, 21. Cambridge University Press, Cambridge, 1991. [14] Grothendieck, A.; Eléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. No. 24, 1965. [15] Grothendieck, A.; Eléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. No. 32, 1967. [16] González-Sánchez, J.; Kirillov’s orbit method for p-groups and pro-p groups. Comm. Algebra 37 (2009), 4476–4488. [17] González-Sánchez, J.; Klopsch, B.; Analytic pro-p groups of small dimensions. J. Group Theory 12 (2009), 711–734. [18] Greenberg, M.J.; Schemata over local rings. Ann. Math. 73 (1961), 624–648. [19] Hrushovski, E.; Kazhdan. D.; Integration in valued fields. Algebraic geometry and number theory, volume 253 of Progr. Math., Birkhäuser Boston, Boston, MA, 2006. [20] Humphreys, J. E.; Conjugacy classes in semisimple algebraic groups. Mathematical Surveys and Monographs 43, American Mathematical Society, Providence, RI, 1995. [21] Huppert, B.; Blackburn, N.; Finite groups. II. Grundlehren der Mathematischen Wissenschaften 242, Springer-Verlag, Berlin-New York, 1982. [22] Isaacs, I. M.; Character theory of finite groups. AMS Chelsea Publishing, Providence, RI, 2006. [23] Jacobson, N.; Lie algebras. Dover Publications, Inc., New York, 1979. [24] Jaikin–Zapirain A.; Zeta function of representations of compact p-adic analytic groups. J. Amer. Math. Soc. 19 (2006), 91–118. [25] Kazhdan, D.; Lecture notes in motivic integration. available at: http://www.math.huji.ac. il/~kazhdan/Notes/motivic/b.pdf. [26] Klopsch B.; On the Lie theory of p-adic analytic groups. Math. Z. 249 (2005), 713–730. [27] Lang, S.; Weil, A.; Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819–827. [28] Larsen, M.; Lubotzky, A.; Representation growth of linear groups. J. Eur. Math. Soc. 10 (2008), 351–390. [29] Liebeck, M.W.; Shalev, A.; Character degrees and random walks in finite groups of Lie type. Proc. London Math. Soc. 90 (2005), 61–86. [30] Lubotzky, A.; Martin, B.; Polynomial representation growth and the congruence subgroup problem. Israel J. Math. 144 (2004), 293–316. [31] Lubotzky, A.; Nikolov, N.; Subgroup growth of lattices in semisimple Lie groups. Acta. Math. 193 (2004), 105–139. [32] Lusztig, G.; On the representations of reductive groups with disconnected centre. Asterisque 168 (1988), 157–166. [33] Margulis, G. A.; Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17. Springer-Verlag, Berlin, 1991. [34] Malle, G.; Testerman, D.; Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011. [35] Nori, M.; On subgroups of GLn (Fp ). Invent. Math. 88 (1987), 257–275. [36] Pas, J.; Uniform p-adic cell decompositions and local zeta functions. J. reine angew. Math 399 (1989), 137–172. [37] Platonov, V.; Rapinchuk, A.; Algebraic groups and number theory. Pure and Applied Mathematics, 139. Academic Press, Inc., Boston, MA, 1994. [38] Prasad, G.; Rapinchuk, A.; Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre, in: Collected papers of John Milnor. V: Algebra (ed. H. Bass and T. Y. Lam), Amer. Math. Soc., Providence, RI, 2010. 66 NIR AVNI, BENJAMIN KLOPSCH, URI ONN, AND CHRISTOPHER VOLL [39] Raghunathan, M.S.; The congruence subgroup problem. Proc. Indian Acad. Sci. Math. 114 (2004), no. 4, 299–308. [40] Serre, J.-P.; Local fields. Graduate Texts in Mathematics, 67. Springer-Verlag, New York, 1979. [41] Serre, J.-P.; Lie algebras and Lie groups. 1964 lectures given at Harvard University. Corrected fifth printing of the second (1992) edition. Lecture Notes in Mathematics, 1500. Springer-Verlag, Berlin, 2006. [42] Tits, J.; Algebraic and abstract simple groups. Ann. of Math. (2) 80 (1964), 313–329. [43] Witten, E.; On quantum gauge theories in two dimensions. Comm. Math. Phys. 141 (1991), no. 1, 153–209. Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston IL 60201, USA E-mail address: avni.nir@gmail.com Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany E-mail address: klopsch@math.uni-duesseldorf.de Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105 Israel E-mail address: urionn@math.bgu.ac.il Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany E-mail address: C.Voll.98@cantab.net