Skip to main content
SpringerLink
Log in

Efficient Computation of BSD Invariants in Genus 2

  • Conference paper
  • First Online:
Book cover Arithmetic Geometry, Number Theory, and Computation

Part of the book series: Simons Symposia ((SISY))

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Artin, Lipmans Proof of Resolution of Singularities for Surfaces. In: G. Cornell, J. H. Silverman (eds), Arithmetic Geometry. Springer, New York, NY, 1986.

    Google Scholar 

  2. B. J. Birch, H. P. F. Swinnerton-Dyer. Notes on elliptic curves. II. J. Reine Angew. Math. 218 (1965), 79–108.

    Google Scholar 

  3. Jonathan W. Bober, Andrew R. Booker, Edgar Costa, Min Lee, David J. Platt, Andrew Sutherland, Computing motivic L-functions, in preparation.

    Google Scholar 

  4. 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.

  5. Spencer Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture. Invent. math. 58, no. 1 (1980): 65–76.

    Article  MathSciNet  Google Scholar 

  6. 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.

    Google Scholar 

  7. Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis. Exp. Math. 15 (2006), no. 4, 385–408.

    Article  Google Scholar 

  8. 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.

    Article  MathSciNet  Google Scholar 

  9. S. Bosch, Q. Liu, Rational points of the group of components of a Néron model. Manuscripta Math. 98 (1999), no. 3, 275–293.

    Article  MathSciNet  Google Scholar 

  10. Nils Bruin, Visualising Sha[2] in abelian surfaces. Math. Comp. 73 (2004), no. 247, 1459–1476.

    Article  MathSciNet  Google Scholar 

  11. N. Bruin, E. V. Flynn, Exhibiting SHA[2] on hyperelliptic Jacobians. J. Number Theory 118 (2006), no. 2, 266–291.

    Article  MathSciNet  Google Scholar 

  12. Edgar Costa and David Platt, A generic L-function calculator for motivic L-functions, available at https://github.com/edgarcosta/lfunctions, 2019

  13. John E. Cremona, Barry Mazur, Visualizing elements in the Shafarevich-Tate group. Experiment. Math. 9 (2000), no. 1, 13–28.

    Article  MathSciNet  Google Scholar 

  14. Brendan Creutz, Improved rank bounds from 2-descent on hyperelliptic Jacobians. Int. J. Number Theory 14 (2018), no. 6, 1709–1713.

    Article  MathSciNet  Google Scholar 

  15. 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.

    Article  MathSciNet  Google Scholar 

  16. David M. Goldschmidt, Algebraic functions and projective curves. Graduate Texts in Mathematics, 215. Springer-Verlag, New York, 2003.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. David Holmes, Computing Néron-Tate heights of points on hyperelliptic Jacobians. J. Number Theory 132 (2012), no. 6, 1295–1305.

    Article  MathSciNet  Google Scholar 

  19. Joseph Lipman, Desingularization of two-dimensional schemes. Ann. Math. 107 (1978), no. 1, 151–207.

    Article  MathSciNet  Google Scholar 

  20. The LMFDB Collaboration, The L-functions and Modular Forms Database. https://www.lmfdb.org.

  21. J. S. Milne, Jacobian varieties. Arithmetic geometry (Storrs, Conn., 1984), 167–212. Springer, New York, 1986.

    Google Scholar 

  22. Pascal Molin, Christian Neurohr, Computing period matrices and the Abel-Jacobi map of superelliptic curves. Math. Comp. 88 (2019), no. 316, 847–888.

    Article  MathSciNet  Google Scholar 

  23. J. Steffen Müller, Computing canonical heights using arithmetic intersection theory. Math. Comp. 83 (2014), no. 285, 311–336.

    Article  MathSciNet  Google Scholar 

  24. Jan Steffen Müller, Michael Stoll, Canonical heights on genus-2 Jacobians. Algebra Number Theory 10 (2016), no. 10, 2153–2234.

    Google Scholar 

  25. Christian Neurohr, Efficient integration on Riemann surfaces & applications. PhD thesis (2018), https://oops.uni-oldenburg.de/3607/1/neueff18.pdf.

  26. B. Poonen, M. Stoll, The Cassels-Tate pairing on polarized abelian varieties. Ann. of Math. 150 (1999), no. 3, 1109–1149.

    Article  MathSciNet  Google Scholar 

  27. Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith. 98 (2001), no. 3, 245–277.

    Article  MathSciNet  Google Scholar 

  28. 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.

    Article  Google Scholar 

  29. 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.

    Google Scholar 

  30. 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.

    Google Scholar 

  31. A. Wittkopf, Algorithms and implementations for differential elimination. PhD dissertation, 2004.

    Google Scholar 

Download references

Acknowledgements

We would like to thank Michael Stoll for sharing his code for the computation of Mordell-Weil groups, and his extensive explanation of this code. Moreover, we thank Edgar Costa for his explanation of the machinery used to compute special values of L-functions. Several anonymous referees are thanked for their useful comments that led to improvements of this article.

Raymond van Bommel has been supported by the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation (Simons Foundation grant 550033).

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Raymond van Bommel

Authors
  1. Raymond van Bommel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raymond van Bommel .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Jennifer S. Balakrishnan

  2. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Noam Elkies

  3. Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI, USA

    Brendan Hassett

  4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Bjorn Poonen

  5. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrew V. Sutherland

  6. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    John Voight

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

van Bommel, R. (2021). Efficient Computation of BSD Invariants in Genus 2. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-80914-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation