Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-17T23:17:17.631Z Has data issue: false hasContentIssue false
  • English
  • Français

A database of genus-2 curves over the rational numbers

Part of: Computational aspects in algebraic geometry Arithmetic algebraic geometry Curves

Published online by Cambridge University Press:  26 August 2016

Andrew R. Booker ,
Jeroen Sijsling ,
Andrew V. Sutherland ,
John Voight  and
Dan Yasaki
Andrew R. Booker
Affiliation:
School of Mathematics, University of Bristol, University Walk, BristolBS8 1TW, United Kingdom email andrew.booker@bristol.ac.uk
Jeroen Sijsling
Affiliation:
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA email sijsling@gmail.com
Andrew V. Sutherland
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email drew@math.mit.edu
John Voight
Affiliation:
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA email jvoight@gmail.com
Dan Yasaki
Affiliation:
Department of Mathematics and Statistics, University of North Carolina at Greensboro, 317 College Avenue, Greensboro, NC 27412, USA email d_yasaki@uncg.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe the construction of a database of genus-$2$ curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated $L$-function. This data has been incorporated into the $L$-Functions and Modular Forms Database (LMFDB).

MSC classification

Primary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 14H45: Special curves and curves of low genus 14Q05: Curves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 235 - 254
Copyright
© The Author(s) 2016 

References

Baker, M. H., González-Jiménez, E., González, J. and Poonen, B., ‘Finiteness results for modular curves of genus at least 2’, Amer. J. Math. 127 (2005) 1325–1387.CrossRefGoogle Scholar
Berry, T. G., ‘On periodicity of continued fractions in hyperelliptic function fields’, Arch. Math. 55 (1990) 259–266.Google Scholar
Birch, B. J. and Kuyk, W. (eds), Modular functions of one variable IV, Lecture Notes in Mathematics 476 (Springer, Berlin, 1975).Google Scholar
Booker, A. R., ‘Numerical tests of modularity’ J. Ramanujan Math. Soc. 20 (2005) no. 4, 283–339.Google Scholar
Bosma, W., Cannon, J., Fieker, C. and Steel, A. (eds), Handbook of Magma functions, edition 2.16 (2010) 5017 pages.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symbolic Comput. 24 (1997) 235–265.Google Scholar
Brumer, A. and Kramer, K., ‘The conductor of an abelian variety’, Compositio Math. 92 (1994) 227–248.Google Scholar
Brumer, A. and Kramer, K., ‘Paramodular abelian varieties of odd conductor’, Trans. Amer. Math. Soc. 366 (2014) 2463–2516.Google Scholar
Bouw, I. I. and Wewers, S., ‘Computing L-functions and semistable reduction of superelliptic curves’, Preprint, 2014, arXiv:1211.4459v4.Google Scholar
Cardona, G., Nart, E. and Pujolas, J., ‘Curves of genus two over fields of even characteristic’, Math. Z. 250 (2005) 177–201.CrossRefGoogle Scholar
Cardona, G. and Quer, J., ‘Field of moduli and field of definition for curves of genus 2’, Computational aspects of algebraic curves, Lecture Notes Series on Computing 13 (World Scientific, Hackensack, NJ, 2005) 71–83.CrossRefGoogle Scholar
Cremona, J. E., Algorithms for modular elliptic curves, 2nd edn (Cambridge University Press, Cambridge, 1997).Google Scholar
Cremona, J. E., ‘The elliptic curve database for conductors to 130000’, Algorithmic number theory: 7th international symposium, ANTS-VII, Lecture Notes in Computer Science 4076 (eds F. Hess, S. Pauli and M. E. Pohst; Springer, Berlin, 2006) 11–29.Google Scholar
Cremona, J. E., ‘Elliptic curve data’, available at https://homepages.warwick.ac.uk/staff/J.E.Cremona/ftp/data/.Google Scholar
Dokchitser, T., ‘Computing special values of motivic L-functions’, Exp. Math. 13 (2004) 137–149.CrossRefGoogle Scholar
Dupont, R., ‘Moyenne arithmético-géométrique, suites de Borchardt et applications’, PhD Thesis, Laboratoire d’Informatique de l’École Polytechnique, 2006, available at http://www.lix.polytechnique.fr/Labo/Regis.Dupont/these_soutenance.pdf.Google Scholar
Farmer, D., Koutsoliotas, S. and Lemurell, S., ‘Varieties via their L-functions’, Preprint, 2015, arXiv:1502.00850v1.Google Scholar
Fité, F., Kedlaya, K. S., Rotger, V. and Sutherland, A. V., ‘Sato–Tate distributions and Galois endomorphism modules in genus 2’, Compositio Math. 148 (2012) 1390–1442.CrossRefGoogle Scholar
González, J., Guàrdia, J. and Rotger, V., ‘Abelian surfaces of GL$_2$-type as Jacobians of curves’, Acta Arith. 116 (2005) 263–287.Google Scholar
González, J. and Rotger, V., ‘Equations of Shimura curves of genus two’, Int. Math. Res. Not. IMRN 2004 (2004) no. 14, 661–674.Google Scholar
González-Jiménez, E. and González, J., ‘Modular curves of genus 2’, Math. Comp. 72 (2002) 397–418.Google Scholar
González-Jiménez, E., González, J. and Guàrdia, J., ‘Computations on modular Jacobian surfaces’, Algorithmic number theory: 5th international symposium, ANTS-V, Lecture Notes in Computer Science 2369 (eds C. Fieker and D. R. Kohel; Springer, Berlin, 2002) 189–197.Google Scholar
Google Inc., ‘Google cloud platform’, available at https://cloud.google.com.Google Scholar
Harvey, D. and Sutherland, A. V., ‘Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time’, Algorithmic number theory (ANTS XI), LMS J. Comput. Math. 17 (2014) 257–273.Google Scholar
Harvey, D. and Sutherland, A. V., ‘Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time II’, Frobenius distributions: Lang–Trotter and Sato–Tate conjectures, Contemporary Mathematics 663 (eds D. Kohel and I. E. Shparlinski; American Mathematical Society, Providence, RI, 2016), 127–148, to appear.Google Scholar
Holt, D., ‘The meataxe as a tool in computational group theory’, The atlas of finite groups: ten years on (Birmingham, 1995), London Mathematical Society Lecture Note Series 249 (Cambridge University Press, Cambridge, 1998) 74–81.Google Scholar
Igusa, J.-I., ‘Arithmetic variety of moduli for genus two’, Ann. of Math. (2) 72 (1960) 612–649.Google Scholar
Johansson, F., ‘Arb: a C library for ball arithmetic’, ACM Commun. Comput. Algebra 47 (2013) no. 4, 166–169.Google Scholar
Kedlaya, K. S. and Sutherland, A. V., ‘Computing L-series of hyperelliptic curves’, Algorithmic number theory: 8th international symposium, ANTS-VIII, Lecture Notes in Computer Science 5011 (eds A. J. van der Poorten and A. Stein; Springer, Berlin, 2008) 312–326.Google Scholar
Lenstra, A. K., Lenstra, H. W. Jr. and Lovász, L., ‘Factoring polynomials with rational coefficients’, Math. Ann. 261 (1982) 513–534.CrossRefGoogle Scholar
Liu, Q., ‘Conducteur et discriminant minimal de courbes de genre 2’, Compositio Math. 94 (1994) 51–79.Google Scholar
Liu, Q., ‘Modèles minimaux des courbes de genre deux’, J. reine angew. Math. 453 (1994) 137–164.Google Scholar
Liu, Q., ‘Modèles entiers des courbes hyperelliptiques sur un corps valuation discrète’, Trans. Amer. Math. Society 348 (1996) 4577–4610.Google Scholar
The LMFDB Collaboration, ‘The L-functions and Modular Forms Database’, available at http://www.lmfdb.org’.Google Scholar
Lockhart, P., ‘On the discriminant of a hyperelliptic curve’, Trans. Amer. Math. Soc. 342 (1994) 729–752.Google Scholar
Merriman, J. R. and Smart, N. P., ‘Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point’, Math. Proc. Cambridge Philos. Soc. 114 (1993) 203–214.Google Scholar
Mestre, J.-F., ‘Construction de courbes de genre 2 à partir de leurs modules’, Effective methods in algebraic geometry, Progress in Mathematics 94 (Birkhäuser, Boston, MA, 1991) 313–334.Google Scholar
The PARI-Group, PARI/GP version 2.7.0 Bordeaux, 2014, available at http://pari.math.u-bordeaux.fr/.Google Scholar
Poonen, B., ‘Computational aspects of curves of genus at least 2’, Algorithmic number theory (ANTS II), Lecture Notes in Computer Science 1122 (ed. H. Cohen; Springer, Berlin, 1996) 283–306.CrossRefGoogle Scholar
Sage Developers, ‘SageMath, the Sage Mathematics Software System (Version 7.2)’, 2016, available at http://www.sagemath.org/.Google Scholar
Sijsling, J., ‘Heuristic determination of endomorphism rings of curves’, 2016, available at https://github.com/JRSijsling/heuristic_endomorphisms/.Google Scholar
Smart, N. P., ‘S-unit equations, binary forms, and curves of genus 2’, Proc. Lond. Math. Soc. (3) 75 (1997) 271–307.Google Scholar
Soundararajan, K., ‘Strong multiplicity one for the Selberg class’, Canad. Math. Bull. 47 (2004) 468–474.CrossRefGoogle Scholar
Sutherland, A. V., smalljac, version 4.1.3, available at http://math.mit.edu/∼drew.Google Scholar
Stein, W. and Watkins, M., ‘A database of elliptic curves—first report’, Algorithmic number theory: 5th international symposium, ANTS-V, Lecture Notes in Computer Science 2369 (eds C. Fieker and D. R. Kohel; Springer, Berlin, 2002) 267–275.Google Scholar
Stoll, M., Genus 2 curve data files, at http://www.mathe2.uni-bayreuth.de/stoll/data/’.Google Scholar
van Wamelen, P. B., ‘Computing with the analytic Jacobian of a genus 2 curve’, Discovering mathematics with Magma, Algorithms and Computation in Mathematics 19 (Springer, Berlin, 2006) 117–135.CrossRefGoogle Scholar