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2021 | OriginalPaper | Chapter

# Efficient Computation of BSD Invariants in Genus 2

Author : Raymond van Bommel

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## Abstract

Recently, all Birch and Swinnerton-Dyer invariants, except for the order of , have been computed for all curves of genus 2 contained in the L-functions and Modular Forms Database [LMFDB]. This report explains the improvements made to the implementation of the algorithm described in [vBom19] that were needed to do the computation of the Tamagawa numbers and the real period in reasonable time. We also explain some of the more technical details of the algorithm, and give a brief overview of the methods used to compute the special value of the L-function and the regulator.
Literature
[Art86]
M. Artin, Lipmans Proof of Resolution of Singularities for Surfaces. In: G. Cornell, J. H. Silverman (eds), Arithmetic Geometry. Springer, New York, NY, 1986.
[BiSw65]
B. J. Birch, H. P. F. Swinnerton-Dyer. Notes on elliptic curves. II. J. Reine Angew. Math. 218 (1965), 79–108.
[BBCLPS]
Jonathan W. Bober, Andrew R. Booker, Edgar Costa, Min Lee, David J. Platt, Andrew Sutherland, Computing motivic L-functions, in preparation.
[vBom19]
Raymond van Bommel, Numerical verification of the Birch and Swinnerton-Dyer conjecture for hyperelliptic curves of higher genus over $$\mathbb Q$$ up to squares. Exp. Math. (2019), doi:​10.​1080/​10586458.​2019.​1592035.
[Blo80]
Spencer Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture. Invent. math. 58, no. 1 (1980): 65–76.
[BHM20]
Raymond van Bommel, David Holmes, J. Steffen Müller, Explicit arithmetic intersection theory and computation of Néron-Tate heights. Math. Comp. 89 (2020), no. 321, 395–410.
[Boo06]
Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis. Exp. Math. 15 (2006), no. 4, 385–408. CrossRef
[BSSVY]
A. R. Booker, A. V. Sutherland, J. Voight, D. Yasaki, A database of genus-2 curves over the rational numbers. LMS Journal of Computation and Mathematics 19, no. A (2016), 235–254.
[BoLi99]
S. Bosch, Q. Liu, Rational points of the group of components of a Néron model. Manuscripta Math. 98 (1999), no. 3, 275–293.
[Bru04]
Nils Bruin, Visualising Sha[2] in abelian surfaces. Math. Comp. 73 (2004), no. 247, 1459–1476.
[BrFl06]
N. Bruin, E. V. Flynn, Exhibiting SHA[2] on hyperelliptic Jacobians. J. Number Theory 118 (2006), no. 2, 266–291.
[CoPl19]
Edgar Costa and David Platt, A generic L-function calculator for motivic L-functions, available at https://​github.​com/​edgarcosta/​lfunctions, 2019
[CrMa00]
John E. Cremona, Barry Mazur, Visualizing elements in the Shafarevich-Tate group. Experiment. Math. 9 (2000), no. 1, 13–28.
[Creu18]
Brendan Creutz, Improved rank bounds from 2-descent on hyperelliptic Jacobians. Int. J. Number Theory 14 (2018), no. 6, 1709–1713.
[FLSSSW]
E. V. Flynn, F. Leprévost, E. F. Schaefer, W. A. Stein, M. Stoll, J. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. Math. Comp. 70 (2001), no. 236, 1675–1697.
[Gol03]
David M. Goldschmidt, Algebraic functions and projective curves. Graduate Texts in Mathematics, 215. Springer-Verlag, New York, 2003.
[Gro82]
B. H. Gross, On the conjecture of Birch and Swinnerton-Dyer for elliptic curves with complex multiplication. Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), 219–236, Progr. Math., 26, Birkhäuser. Boston, Mass., 1982.
[Hol12]
David Holmes, Computing Néron-Tate heights of points on hyperelliptic Jacobians. J. Number Theory 132 (2012), no. 6, 1295–1305.
[Lip78]
Joseph Lipman, Desingularization of two-dimensional schemes. Ann. Math. 107 (1978), no. 1, 151–207.
[LMFDB]
The LMFDB Collaboration, The L-functions and Modular Forms Database. https://​www.​lmfdb.​org.
[Mil86]
J. S. Milne, Jacobian varieties. Arithmetic geometry (Storrs, Conn., 1984), 167–212. Springer, New York, 1986.
[MN19]
Pascal Molin, Christian Neurohr, Computing period matrices and the Abel-Jacobi map of superelliptic curves. Math. Comp. 88 (2019), no. 316, 847–888.
[Mül14]
J. Steffen Müller, Computing canonical heights using arithmetic intersection theory. Math. Comp. 83 (2014), no. 285, 311–336.
[MüSt16]
Jan Steffen Müller, Michael Stoll, Canonical heights on genus-2 Jacobians. Algebra Number Theory 10 (2016), no. 10, 2153–2234.
[Neu18]
Christian Neurohr, Efficient integration on Riemann surfaces & applications. PhD thesis (2018), https://​oops.​uni-oldenburg.​de/​3607/​1/​neueff18.​pdf.
[PoSt99]
B. Poonen, M. Stoll, The Cassels-Tate pairing on polarized abelian varieties. Ann. of Math. 150 (1999), no. 3, 1109–1149.
[Sto01]
Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98 (2001), no. 3, 245–277.
[Suth19]
A. V. Sutherland, A database of nonhyperelliptic genus 3 curves over Q, Thirteenth Algorithmic Number Theory Symposium (ANTS XIII), Open Book Series 2 (2019), 443–459. CrossRef
[Tate66]
J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki, Vol. 9 (1964–1966), Exp. No. 306, 415–440, Soc. Math. France, Paris, 1995.
[vWam06]
Paul B. van Wamelen, Computing with the analytic Jacobian of a genus 2 curve. Discovering mathematics with Magma, 117–135. Algorithms Comput. Math., 19, Springer, Berlin, 2006.
[Witt04]
A. Wittkopf, Algorithms and implementations for differential elimination. PhD dissertation, 2004.