Title Formal groups and zeta-functions

Author Honda, Taira

Citation Osaka Journal of Mathematics. 5(2); 199-213
Issue Date 1968-12
ISSN 0030-6126
Textversion Publisher
The OJM has been digitized through Project Euclid platform
Relation
http://projecteuclid.org/ojm starting from Vol. 1, No. 1.

Placed on: Osaka City University

Osaka City University

Honda, T.
Osaka J. Math.
5 (1968), 199-213

FORMAL GROUPS AND ZETA-FUNCTIONS

TAIRA HONDA

(Received July 8, 1968)

Let F(x, y) be a one-parameter formal group over the rational integer ring
Z. Then it is easy to see that there is a unique formal power series f(x)—
2 n~lanxn with an^Z, a^=\ satisfying

and that f'(x)dx= 2 o,nxn~ldκ is the canonical invariant differential on F. Let

»=1

Cl be an elliptic curve over the rational number field Q, uniformized by automor-

phic functions with respect to some congruence modular group T0(N). In the
language of formal groups results of Eichler [3] and Shimura [14] imply that a
formal completion C1 of Cl (as an abelian variety) is isomorphic over Z' to a
formal group whose invariant differential has essentially the same coefficients as
the zeta-f unction of Q.
In this paper we prove that the same holds for any elliptic curve C over Q
(th. 5). This follows from general theorems which allow us explicit construction
and characterization of certain important (one-parameter) formal groups over
finite fields, p-adic integer rings, and the rational integer ring (th. 2 and th. 3).
The proof of th. 5 depends only on the fact that the Frobenius endomorphism of
an elliptic curve over a finite field is the inverse of a zero of the numerator of the
zeta-function, and implies a general relation between the group law and the zeta-
function of a commutative group variety. In fact it is remarkable that the p-f actor
of the zeta-function of C for bad p also can be given a clear interpretation from our
point of view (cf. th. 5). Moreover, we prove that the Dirichlet L-function with
conductor D has the same coefficients as the canonical invariant differential on
a formal group isomorphic, over the ring of integers in Q(\/ D), to the algebroid
group x+y+\/T) xy (th. 4). In this way the zeta-function of a commutative
group variety may be characterized as the L-series whose coefficients give a normal
form of its group law.

1. Preliminaries
Let R be a commutative ring with the identity 1. We denote by R{x},
200 T. HONDA

R{x, y}, etc. formal power series rings with coefficients in R. Two formal power
series are said to be congruent (mod deg n) if and only if they coincide in terms of
degree strictly less than n. A one-parameter formal group (or a group law) over
R is a formal power series F(x, y)^R{x, y} satisfying the following axioms:

(i) F(z, 0) = F(0, z) = z

(ii)

If F(x, y)=F(y, x) moreover, F is said to be commutative. Let G be another

group law over R. By a homomorphism of F into G we mean a formal power
series φ(x)^R{x} such that <p(0)=0 and φ°F=G°φ, where we have written
G(φ(x\ φ(y))=(G°φ)(x, y). If φ has the inverse function φ~\ φ~l is also a
homomorphism of G into F. In this case we say that G is (weakly) ίsomorphίc
to F and write φ: F~G. If there is an isomorphism φ of F onto G such that
φ(x) = x (mod deg 2), we say that G is strongly isomorphίc to F and write φ: F^
G. If G is commutative, the set Hom#(F, G), consisting of all the homomorphims
of F into G over R, has a structure of an additive group by defining (cp^-
= G(φ1(x), φ2(x)) for φly φ2&HomR (F, G). In particular End^ (F) (^
(F, F)) forms a ring with the identity [1] (x)=x. We call [n]F the image of n^
Z under the canonical homomorphism of Z into EndR(F).
Writing A=R{x}, we denote by ®(^4 R) the space of ^-derivations of A.
It is a free ^4-module of rank 1 and is generated by D=d/dx. We denote by
®*(^4 R) the dual ^4-module of ®(^4 R) and call its element a differential of
A. ¥orf<=A the map D-*Df of ®(^4; #) into ,4 defines a differential, which
we denote by df. A differential of the form df with f^A is called an exact
differential. It is easy to see that dx is an ^4-basis of 3)*(^4; R) and df=(Df)dx
for any f^A. Let ω=^(x)dx be a differential of A and let <£>(#) e .4 with
<p(0)=0. Then Λjr(φ(x)) dφ(x) is again a differential. We denote it by φ*(ω).
The map ^>* is an /?-endomorρhism of ®*(^1 R). Let F(x, y) be a (one-
parameter) formal group over R. Introducing a new variable ty F is considered
a formal group over R{t}. Define the right translation Tt of F by Tt(x)=F(x, t).
A differential ω of A is said to be an invariant differential on F if and only if
7γ*(ω)=ω. The set of all the invariant differentials on F forms an Λ-module.
We denote it by ®*(F; R).

Proposition 1. Let F(x, y) be a one-parameter formal group over R. Put

ψ(z)=(-j-F(Q9 z)\ * and ω=ψ(x)dx. Then we have ψ (0)=l and^^(F\ R) is a
\ox /
free R-module of rank one generated by ω.
r\

Proof. Since F(xy y) = x+y (mod deg 2), we have ^--^(0, z)=l (mod deg
1). Hence ψ(ar) is well-defined and ψ(0)=l, A differential η=\(x)dx of A
 FORMAL GROUPS AND ZETA-FUNCTIONS 201

is invariant on F if and only if \(x)dx=\(F(x, z))—F(x, z)dx, or

ox

(*) = \(F(x, *)) F(*, z) .

dx
From (1) we have
x(θ) = λ(*)/r(o,
OX
or
(2) λ(*) =
Define an Λ-homomorphism Φ of ®*(F; R) into R by Φ(ι?)=λ(0). By (2) Φ
is injective. Now differentiating F(u, F(v, w))=F(F(u, v), w) relative to u, we
obtain
(u, F(v, w)) = j-F(F(u, υ), w)j-F(u, v),
dx dx
and then
F(0, F(v, «0) = * («, *v)-F(Q, v ) ,
OX OX OX

or

Now (3) implies that ψ(#) satisfies (1). Therefore ω belongs to 2)*(F; R) and
is clearly its /?-basis.
We shall call this ω the canonical invariant differential on F.
Proposition 2. Let F be a one-parameter formal group over a Q-algebra R.
Then we have F(x, y)^=ίx-\-y over R.
Proof. As R is a Q-algebra, all the differentials of A are exact. Let ω=
df(x) with/(#)ΞΛ?(mod deg 2) be the canonical invariant differential on F. Then
we have df(F(x, t))=df(x), i.e. f(F(x, t))-f(x)€ΞR{t}. Put f(F(x, t))=f(x)+g(t).
Then we have f(F(Q, t))=Q+g(t), or g(t)=f(t). Since f(x) is inversible, this
completes the proof.
Prop. 2 was proved in Lazard [5] in an alternative way. More generally we
can prove that a commutative formal group of arbitrary dimension over a Q-
algebra is strongly isomorphic to the vector group of the same dimension.
Now let R be an integral domain of characteristic 0 and let K be the fraction
field of R. We note that, if φ(x)^R{x} satisfies the functional equation φ(x+y)
=φ(x)-}-φ(y), φ(χ) must be of the form ax with a^R. Let F and G be group
laws over /?, let φ e Hom^(F, G) and let c(φ) be the first-degree coefficient of <p,
 202 T. HONDA

The additive map c:φ-*c(φ) of HomR(F, G) into R, which is a unitary ring-

homomorphism in the case F=Gy is injective, because F (resp. G)^x-\-y over K
(cf. Lubin [6]). In particular the seήes f(x) ^ K{x} such that f(x)=x (mod deg 2)
and F(#, y)=f~\f(x)+f(y)) is uniquely determined by F. For this / and for
e put [Λ]F(*)=/~^Λ/(ΛO) It *s clear tnat MFe End^F) if and only if

We now consider formal groups over a field k of char act eristic p>0.

Lemma 1. Let F and G be group laws over k. If φ<^Homk(F, G) and if

], there is q=pr such that φ(x) = axq [mod deg (#+!)] with 0ΦO. Moreover
φ(x) is a power series in xq .

Proof. See Lazard [5] or Lubin [6],

If [p]F(x) = axq [mod deg (#+!)] with 0ΦO and q=ph, h is called the height
of F. If [ρ]F=Q, then the height of F is said to be infinite (Lazard [5]). We
denote by h(F) the height of F. It is easy to see that, if A(F)ΦA(G), then
Hom^F, G)=0.
Now it is well known that k{x} has the structure of a topological ring if we
take powers of its maximal ideal as a basis of neighbourhoods at 0. Endowed
with the topology induced by it, Hom^F, G) (resp. Endfc(F)) becomes a complete
topological group (resp. ring) (Lubin [6]). It is clear that End^F) has no zero-
divisor. Moreover it is easy to see that, if h(F)<o°, the homomorphism
n->[ri\F of Z'mto End^(F) is injective and this imbedding is continuous relative
to p-adic topology of Z. Since End^(F) is complete, this extends to an imbed-
ding of the />-adic integer ring Zp into Endk(F). In this way End^F) is a
Z^-algebra and Hom^(F, G) is a Z^-module.
The following theorem is fundamental in the theory of one-parameter formal
groups over a field of positive characteristic.

Theorem 1. (Lazard [5], Dieudonne [2] and Lubin [6].)

(i) For every h(l^h^°°) there is a formal group of height h over the prime
field of characteristic p>0.
(ii) Let k be an algebraically closed field of characteristic p>Q. If F and G
are group laws over k and if h(F)=h(G), then F~G over k. Moreover, if h(F)
=h(G)=ooy then F^G over k.
(iii) Let k be as in (ii) and let F be a group law over k. If h=h(F)<°°,
Endk(F) is the maximal order in the central division algebra with invariant l/h
overQp.
Later we shall reprove (i) and (iii) as applications of our results in 2.

2. Certain formal groups over finite fields and p-adic integer rings
Let R be a complete discrete valuation ring of characteristic 0 such that the
 FORMAL GROUPS AND ZETA-FUNCTIONS 203

residue class field k=RI\n is of characteristic ^>>0, where m denotes the maximal
ideal of R. For a group law F over R we obtain a group law over k by reducing
the coefficients of F mod m. We denote it by F*. If G is another group law
over 7?, we derive the reduction map *: Hom^F, G)^Homk(F*, G*). The
following two lemmas are due to Lubin [6].

Lemma 2. The map c : HomR(F, G)-*R is an isomorphism onto a closed

subgroup of R.
This is Lemma 2.1.1. of [6].

Lemma 3. If h(F*)<o°, the reduction map *: Hom^F, G)

G*) is injective.
This is lemma 2.3.1. of [6].
From now on until the end of 2 we denote by o the integer ring in an exten-
sion field K of Qpy of finite degree n, and by £ the maximal ideal of o. Let e and
d be the ramification index and the degree of p respectively. The residue
classs field o/p is the finite field Fq with q elements, where q=pd. The following
two lemmas play essential roles in our further investigation.

Lemma 4. Let π be a prime element of o. For any integers z>2^0,

and m^l we have
mp
^ (mod J>) .

Proof. It suffices to prove our lemma for a=m=l. We have to prove

for 1
={=PV '

This is trivial if i~^v. Assume i<v. Let p* \ /!, but pμ+1 Xi\. Then we see

μ = [ilp\ + [ilf\+...<ilp+ilp*+... = i/(p-l)^i.

Hence we have

(f y-v = (p*-\)-(p'-i+\ypili\ =0 (mod/,) ,

and a fortiori (4).

The following lemma is a trivial generalization of [7], lemma 1.

Lemma 5. Let π be a prime element of o and let a^>l be an integer.

Let f(x) and g(x) be power series in o{x} such that
( 5) f(x) =g(x) = πx (mod deg 2) and f(x)=g(x) = xqa (mod p) .

Moreover, let L(zly •••, zn) be a linear form with coefficients in o. Then there exists
a unique power series F(zv •••, zn) with coefficients in o such that
204 T. HONDA

F(*ι> — i z») = £(#!, •••, zn) (mod deg 2)

( 6) and
f(F(*ί9 .,zn)) =

Proof. See Lubin-Tate [7]. Note that F is the only power series with
coefficients in any overfield of o satisfying (6).
Denote by O the ring of integers in the maximal unramified extension of
K. We are now ready to prove the following:

Theorem 2. Let π be a prime element of o and let a^l be an integer.

ΛV
Put /(A?)=Σ 7Γ~V and F(xy y)=f~1(f(x)+f(y)). Then we have the following:
v=o
(i) F is a group law over o and Endo(F) is the integer ring of the unramified
extension of K of degree a.
(ii) F* is a group law of height an over Fq . Denoting by ξF* the q-th
power endomorphίsm of F* (i.e. ξF*(x)=x9), we have

(7) [*]£=&..
(iii) If G is another group law over o such that [τr]GeEnd0(G) and such
that [7r]g=£c*, then F^G over o.

Proof. We define u(x)^K{x} by

( 8) [π]F(x) = f-\πf(x}) = tf*+πu(x) .

We shall prove u(x)^o{x}. From (8) we have

πf(X)=f(X«a+πu(x)),
πx+ Σ π^x9^^ = x9"+πu(x)+ f] π~\xqa ^-πu(x)\qa"
v=o v=ι
and

( 9) π(x-u(x)) = Σ [π-\x

Put u(x)=x-\-^] bjX* and assume b29 •••, δ^.^o. Since bk is written as a poly-
ί=2

nomial of b2y •••, bk^ by (9), we have bk^o by applying lemma 4 to (9). This
proves u(x) e O{ΛJ} .
This being proved, we can apply lemma 5 to [/τ]F(#) as is seen from (8).
First F(xy y)^o{x, y} follows from \π\F°F=F°\π\F by lemma 5. The equality
(7) follows directly from (8). Now putp=8πe. Then 8 is a unit in o. We have

hence, by (7),
 FORMAL GROUPS AND ZETA-FUNCTIONS 205

[p]F* = (automorphism of F*)°ξFf*

Since ξa/*(x)=χpda\ we have h(F*)=dae=an, which completes the proof of (ii).

Let G be as in (iii). By prop. 2 there is φ(x)^K{x} with φ(x) = x (mod deg 2)
such that φ°F=G°φ. Then we have φo[π\F= [π]G°φ. Hence φ has coefficients
in o by lemma 5.
It remains to determine Endo(F). Let w be a primitive (qa— l)-th root of
unity in £). By definition of f(x) we have f(wx)=wf(x) and so F(wx, zvy)=
wF(xy y). Hence we have wx= [w\F\x\ eEndo(F). This implies that the fraction
field L of Endo(^) contains the unramified extension of Qp of degree ad. More-
over, since [π]F e End£)(.F), the ramification index of L/Qp is a multiple of e.
Thus we have [L: Qp]^ade=an. On the other hand, as h(F*)=any we have
[L: Qp]^an by th. 1, (iii) and by lemma 3. Hence we have [L: Qp]=an. Since
Zp[wy π] is the integer ring of L, this proves (ii) and completes the proof of th. 2.
The existence of a formal group F with the properties (i), (ii) in th. 2 was
proved by Lubin ([6], th. 5.1.2.). But his construction of F is not explicit
as ours.
Corollary. Let F be a formal group over Zp such that h(F*)=l. Then we
can find a prime element π of Zp such that [π]F (x)=xp. The map: F-*π gives a
bijection Φ: [strong isomorphism classes of formal groups F over Zp such that
h(F*)=l}-*{prime elements of Zp}.

Proof. Since λ(F*)=l, the map *: Endz/,(F)->EndF/,(F*) is bijective by

th. 1, (iii). As ξF*(x)=xp ^EndFp(F*), this proves the first assertion. The
injectivity of Φ follows from th. 2, (iii) and the surjectivity from th. 2, (ii).
We now prove th. 1, (iii) assuming th. 1, (ii). Applying th. 2 to o=Zp and
f(x)= Σ/>~ V #* AV > we obtain a group law F* over Fp, of height h. Let k be the
algebraic closure of Fp. Since Endk(F*) contains \w\% and ξF*, EndΛ(F*) contains
the maximal order Mh in the central division algebra Dh of rank h2 over Qp, and
invariant l/h. (For details see [6], 5.1.3.) We shall prove EndΛ(jP^)=MΛ. In the
following we write ξ instead of ξF* for simplicity. Let nh be the integer ring in
the unramified extension of degree h over Qp and let S be a system of representa-
tives of Mh modulo its maximal ideal. For β^S, we write [β] instead of [β]F
for brevity. Then we have [β](x) = β*x (mod deg 2). Let φ be any element of
End^F*) and let φ(x) = aQx (mod deg 2). Comparing the r-th degree coefficients
of φ°[p]f= [P]F°<P, where r=ph, we have α0^α0r, i.e. a0^Fr. Hence we can
find β0^S such that (φ—[β0]) (x) = Q (mod deg 2). Then, by lemma 1, there is
φ1 eEnd k (F*) such that φ— [βo^φ^ξ Applying the same argument to φ19
we obtain β^S and φ2eEndk(F*) such that φ1—[βι]=φ2°ξ By repeating
the same procedure n-times we derive β0, βly •••, βn_1^S and φ19 φ2, •••, φn^
*) such that φi—[βi}=φi+^ξ for O^i^n— 1, where φ^=φ> Then
206 T. HONDA

we have

Hence the series [β0] + [βι]ξ~\-----\-[βn-ι]%"~l+ " converges and coincides with
φ. Since [/3, ]eMA, this proves

REMARK. Formal groups F* constructed in th. 2 do not exhaust all the

formal groups over finite fields (cf. Serre [13], p. 9).

3. Certain formal groups over Z

We now give explicit global construction of certain formal groups over Z.
The method is based on lemma 4 and lemma 5 as in 2.

Lemma 6. Let p be a prime number and let a^ a2, •••, any ••• be rational
integers satisfying the following conditions:
( i ) If n—p"m with pXm, then an=ap^am
(ii) β 1= =l. p/ap.

= 0 for v^Q .

Let π be the prime element of Zp satisfying the equation

(10) X2-apX+p = 0 .

Put f(x)=^n~1anxn and F(x, y)=f-\f(x)+f(y)). Then we have F(x,

{x] and [π]F(x) = xp (mod p).

Proof. By HenseΓs lemma and by the assumption pXap the equation (10)
has solutions in Zp. Let π be the other root of (10). It is a unit in Zp. Since

apv+2—(π+π')apv+ι+ππ'ap » = 0 ,
we have

(11) ap-»+2— πf apv+\ = π(apv+ι — π' a^) for z^^O .

Define u(x)^Qp{x} by

(12) [π]F(x) =f~l(πf(x)) = x*+πu(x) .

The point of the proof is to prove u(x)^Zp{x} as in th. 2. From (12) we obtain

7τ Σ n~lanxn = xp+πu(x)+ f j n-1an(xp+πu(x))M ,

»=1 »=2

or

(13) π(x—u(x)) = xp+ Σ n'1an(xp+πu(x))H—π Σ ιΓlanxM ,

—
 FORMAL GROUPS AND ZETA-FUNCTIONS 207

χi
Put u(x)=Σbi > where ^=1. Assuming ό2, •••, bk^^Zpy we shall prove
bk^Zp. By lemma 4 we have
1 p
*-ι l pn
n~ (x -\-π'Σ biX*)" = n~ x (mod^>).

Hence by (13), we have only to prove that the k-th degree coefficient ck in
1 pn 1 n
(14) Σ n- anx -π^n- anx
n=i n=2

is a multiple of p. If p/^k, this is clear. Assume k=p*m with z>^l,

We have

or

(15) ck=p^m-lamπ(π/a^-ι-ap^ .

Applying (11) to (15) repeatedly we have

ck = p-^m^am7^(πfa^-a^
= -p-*m->amπ^
= 0 (mod p) .

This proves bk^Zp and by induction we see in fact u(x)^Zp{x}. The fact
F(x, y)^Zp{x, y] follows from this by Lemma 5. (cf. The proof of th. 2)

Lemma 7. Let p be a prime number, let 6= + 1 or — 1, and let A^>1 be

an integer. Let aly a2, •••, any ••• be rational integers satisfying the following condi-
tions :
(i) If n=p"m with pXm, then an=ap*am.
(ii) a,= l. ap= ••• =aph-ι=0.
apv+tι=6ph-ιap'» for v^>0.

Put f(x)=^n-lanxn and F(x, y)=f~l(f(x)+f(y)). Then we have Fix, j)eΞ

»=1
ph
Zp{x,y} and [8p]F(x) = x (mod p).

Proof. Repeat the same reasoning as in the proof of lemma 6. The point
is to prove u(x)^Zp{x}, where u(x) is defined by [8p]F(x)=xph-\-pu(x). The
details will be left to the reader.

Theorem 3. Assume that to every prime number p there is given a local

L-series Lp(s) of the type :
 208 T. HONDA

(a) Lp(s)=l,
(b) Lf(s)=(l-atp-+p1-^-1 with ateZ, pXap,
or
(c) L,(ί)=(l-£y'-*•)-' with εp= + l or -1, h=hp^\.

Define the global (format) L-series £(ί)=Σ ann~s by L(s)=ΐl Lp(s) and put
w=l p
l n 1
/(*HΣ n~ anx . Then the formal group F(x, y)=f~ (f(x)+f(y)) has coefficients
in Z. Denote by F* the reduction of F mod p. Then we have :
Case (a): F^x-\-y over Zp.
Case (b): h(F*)=\ and the p-th power endomorphism of F* is a root of the
equation
X*-apX+p = 0 .

Case(c): h(F*}=h and [£pp]F(x) = xph (modp).

Proof. If Lp(s)= 1, the coefficients off(x) are ^-integral and we have F(x, y)
z&x+y over Zp. If Lp(s) is of type (b) (resp. (c)), it is easily verified that the
sequence aly a2y •••, an, ••• satisfies the assumptions of lemma 6 (resp. lemma 7).
Therefore the coefficients of F(xy y) are ^-integral for every p. This proves
F(x, y)^Z{xy y}. The other assertions of our theorem follow from lemma 6 and
lemma 7.
The following proposition is useful in the study of algebroid commutative
formal groups over Q.

Proposition 3. Let p be a prime number and let o be the integer ring of the
quadratic unramίfied extension of Q^ v
Put f1(x)=^p~^xp^y /2(#)=Σ (—pY^xp^
v=o v^o
and Fi(x, y)=f7l(fi(x)+fi(y)) for i=l, 2. Then we have the follwoίng:
(i) Ff~Ff over Fp*, but Ff^Ff over Fp. If p is odd, then F^F2
over o.
(ii) Let F be a group law over Zp such that F*(x, y)~x+y+xy over Fp*.
Then we have either F^F1 or F^F2 over Zp according as F*(x,y)~~x-}-y-\-xy
over Fp or not.

Proof. By th. 3 F, (i=l, 2) has coefficients in Z and [p]Fί(x) = [—p]F2(x)

p
= x (mod p). Let k be the algebraic closure of Fp. Since h(Ff)=h(Ff)=l,
there is an inversible series φ(x)^k{x] such that φ°Ff=Fξ°φ by th. 1, (ii).
Then we have 9>°[/>2]!^=[^*200>, i.e. φ(xp2) = φ (x)p2. This implies φ(x)
{x} and Ff~~Ff over Fp*. If φ(x)^Fp{x}> we should have
 FORMAL GROUPS AND ZETA-FUNCTIONS 209

and then

a contradiction. Hence Ffs&Ff over Fp. If p is odd, o contains the primitive

(p2— l)-th root of unity and there is zυ^o such that wp"1= — l. Then we have
wp*=( — iγw. Hence f1(wx)=wf2(x) and then F^wXj wy)=wF2(x, y), which
proves (i). Now the p-th power endomorphism of F* conies from an endomor-
phism of F9 say [π]Fy since λ(F*)=l. As the />-times endomorphism of the
multiplicative group x+y+xy over Fp is (l-\-x)p—l=xp, we have Ff(x, y)~x-\-
y-\-xy over Fp by th. 1, (ii) and so F*~x-\-y-\-xy~Ff over Fp*. Let ψ be an
inversible element of Fp*{x} such that tyoF*=Ff°ty. Then

which implies π2=p2. Then by th. 2, (iii) we have F^Fl or F^F2 over Z^
according as π=p or — />, i.e. according as F*~~x-\-y-}-xy or not.

4. Group laws and zeta-functions of group varieties of dimension

one
We now interprete zeta-functions of certain commutative group varieties
from our point of view. Let F(x, y) be a group law over Z. Then there is unique
f(x)£ΞQ{x} such that /(*) = # (mod deg 2) and F(x, y)=f~1(f(x)+f(y)) (cf. 1).
It is clear that df(x)=f'(x)dx is the canonical invariant differential ω on F. Let
00 00

/'(Λ?)=Σ anxn~l and define a (formal) L-series L(ί) by L(s)=Σ ^»^~s If each
one of F, /, ω and L(s) is given, the rests are uniquely determined from it.

Theorem 4. Let K be a quadratic number field, let o be the integer ring of

00
/D \
^ 0m/ let D be the discriminant ofJ K. Then the Dίrίchlet L-function
J Y] — }n~s
ίfci V n )
is obtained from a group law G(x,y) over Z. Moreover, let F(xJy)=x-\-y+\/'Dxy.
Then we have F^G over o.

Proof. Let %(w)— ( — ) be the Kronecker symbol and define

\n /
(16) P(u)= Π (1-Γ«), where ?=exp (2τt^^\l\D\) .
a mod D
X(α)=l
It is easy to see P(u)Eϊo[u]. Let σ be the non-trivial automorphism of K and
put

(17) φ(u) = (P*(u)-P

We have only to prove that φ(u)=u-\----^o{u} and

210 T. HONDA

(18) dφ(u)l(\+VDφ(u)) = Σ K(rί)un^du ,

since dx/(\JΓ\/~Dx) is the canonical invariant differential on F. We recall

(19) Σ X(r)ζ"r = X(/zX/Z) for any n<=Z

r mod D

(Gauss sum). The first-degree coefficient of φ(u) is

(- b mod
Σ D ξ"+ a mod
Σ D niVD
5C(δ) = -l X(fl) = l

= (r mod£>
Σ X(r)r)/v/Π=l

by (19). Let αt be the ί-th degree coefficient of Pσ—P. We shall prove #,- = ()
(mod x/Z)). Since (Pσ-P)σ=-(Pσ-P), α, is of the form c,VZ> with 2r,eZ.
If Z) is odd, we have at once c^Z. If D is even, we have D = Q (mod 4). In
this case we can easily check

X(r+D/2) = - X(r) for any r EΞ 2Γ

and so {ξ- Λ |β mod Z>, %(α)=l} coincide with {—ξ b \ b mod D, X(b)= — l} as a
whole. Hence af=0 or twice an integer according as / is even or odd. This
shows Ci^Z and φ(u)^o{u}. Let us compute dφ(u)/(l-}-\/J)φ(u)). We have

dφ(u) =

V!_^>

VD Ί
= N/ZT'-P"'1!] Σ X(r)ζ'"'uM-1du
»=i r mod D

= P"'1!] X(n)u"-ldu (by (19)).

Hence we have
00

7 / \ .i^ x [ /C(71)U
v; ClU
dφ(u) _ jfcj

This completes the proof of our theorem.

Now the Dirichlet L-function L(s, %) has an Euler product of the form
Π(l—£ p p~ s )~ l where ^=%(ί>). By th. 3 8p is uniquely determined by the group
P
law F. From this point of view L(s, %) can be characterized as the L-series
attached to a normal form over Z of the algebroid group F. The Euler product
 FORMAL GROUPS AND ZETA-FUNCTIONS 211

implies that the group law F is "the direct product" of group laws over Zp's
attached to p-f, actors of L(s, %).
Quite the same holds for elliptic curves over Q. In the following we mean
by an elliptic curve an abelian variety of dimension one. Let C be an elliptic
curve over Q. Neron [10] shows that there is an essentially unique (affine)
model for C of the form

(20) Y2+\XY+μY = X3+aX2+βX+j

where λ, μ, a, β, γ are integers and the discriminant of the equation (18) is as

small as possible. For this model Cp=C mod p is an irreducible curve for every
prime number p. Then local L-series Lp(s) of C are defined as follows.
( I) If Cp is of genus 1, we put

where l—apU+pU2 is the numerator of the zeta-function of Cp.

(II) If Cp has an ordinary double point, we put 8p=-\-l or —1 according
as the tangents at the double point are rational over Fp or not and write

(III) If Cp has a cusp, we put

Lp(s) = 1 .

In case (II) the reduction of the group law of C is isomorphic to the mul-
tiplicative group over Fpz and is isomorphic to it over Fp if and only if 6p=-\-l.
In case (III) the reduction of the group law of C is the additive group ([10], Chap.
Ill, prop. 3).
Now, we take t=X/Y as a local parameter at the origin. By [15], Chap.
Ill, prop. 4 t is a local parameter at the origin of Cp for every p. Writing down
the group law of C as a formal power series relative to the variable t, we obtain a
formal group F(ocy y) over Z. (The fact F(x, y) e Z{x, y} can be verified also by
direct computation.) We shall call a formal group over Zy strongly isomorphic
to this F over Z, a formal minimal model for C over Z.

Theorem 5. Let C, Cpy Lp(s) and F be as above. Let S be any set of prime
numbers which does not contain p=2 or 3, if Cp has genus one and ap=±p, and
put Zs={\ (ZPΠQ). Write ΠLp(s)=Σ ann~s, g(x)=Σ n~λanxn and G(x9 y)
p£=S p^S n=l n=ι
1 x
=S~ (s( )+S(y))' Then G(x, y) is a formal group over Z and F^G over Zs.

Proof. If Cp has genus one and p \ ap, we see easily ap=0 or ap=±p with
p=2 or 3 by Riemann hypothesis | ap \ <2^/~p . The latter cases being excluded,
 212 T. HONDA

we can apply th. 3 to Π Lp(s) and obtain G(x, y)^Z{x, y}. In order to show
F^G over Zs, we have only to prove F^Cover Zp for every p^S, since a
power series φ(x) such that φ(x) = x (mod deg 2) and φ°F=G°φ is unique.
If Cp has genus one for p^Sy then F^G over Z^ by th. 3 and th. 2, (iii), since
X2—apXJrp is the characteristic polynomial of the ^>-th power endomorphism
of Cp. In case (II) F mod^> is isomorphic to the multiplicative group x-\-y-\-xy
over Fpz and isomorphic to it over Fp is and only if £p= +1. Hence we have
F^G over Zp by prop. 3, (ii), by th. 3 and by th. 2, (iii). In case (III) it is clear
F^G over Zp. This completes our proof.

REMARK. It seems that the assumption on S in th. 5 would be superfluous.

But I have not been able to get rid of it.

Corollary 1. Notations being as in th. 5, assume that ap^=±pfor p=2, 3.

Then the formal group attached to the zeta-functίon L(s\ C)=TlLp(s) of C has
coefficients in Z and is a formal minimal model for C.

Corollary 2. Let C and C' be elliptic curves over Q and let Shea set of primes
satisfying the assumption in th. 5 for each curve. Then formal minimal models of
C and C' are isomorphic over Zs, if and only if p-factors of L(s\ C) and L(s\
C') coincide for every

Corollary 3. Let notations be as in th. S. If Cp has genus one for

ap mod p is the Hasse invariant of Cp.

Proof. Take f(x)^Q{x} such that f ( x ) = x (mod deg 2) and F(x, y)=
χ
f~*(f( }^~f(y)}' Then/'(Z)Λ is the canonical invariant differential on F, i.e. the
^-expansion of an differential of the 1st kind on C. Hence our assertion follows
from definition of Hasse invariant and from th. 5.

REMARK. Coroll. 3 is a special case of th. 1 of Manin [9]. So his theorem

is suggestive for generalization of th. 5 to an abelian variety of higher dimension
over an algebraic number field.

Corollary 4. Let C be an elliptic curve over Q and assume ap=0 for a prime
number p. Denote by o the integer ring of the quadratic unramίfied extension of Qp.
Then C has formal complex multiplications over o, i.e. End0(F)=o.

Proof. Let H be the formal group over Z attached to the L-s-ries (1 +

1 2 1 pz
p - *)- . We have H(x, y)=h-\h(x)+h(y)) where A(*)=Σ (-p)^x \ If
ap~Q, then F^H over Zp by th. 5, and our assertion follows from th. 2, (i).

REMARK. Existence of elliptic curves, which have no complex multiplication

but have formal complex multiplications over p-adic integer rings, was proved by
 FORMAL GROUPS AND ZETA-FUNCTIONS 213

Lubin-Tate [8]. But they did not give an explicit example. Our result has a
meaning in the study of /-adic Lie groups attached to elliptic curves overQ.
(cf. Remark on p. 246 of Serre [12].)
There are some questions concerned with our results. How can we gen-
eralize th. 4 to more general L-functions ? Let F and G be as in th. 5 with S=
the set of all the prime numbers. What is the power series φ(x) e Z{x] such that
φ(x) = x (mod deg 2) and F°φ^φ°Gϊ How can we generalize th. 5 to an abelian
variety of higher dimension over an algebraic number field ?

OSAKA UNIVERSITY.

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