Skip to main content Accessibility help
×
Home

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

  • Jennifer S. Balakrishnan (a1), Wei Ho (a2), Nathan Kaplan (a3), Simon Spicer (a4), William Stein (a5) and James Weigandt (a6)...

Abstract

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$ -Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$ , ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$ , ordered by height, in which we compute ranks and $2$ -Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
      Available formats
      ×

Copyright

References

Hide All
1. Balakrishnan, J. S., Ho, W., Kaplan, N., Spicer, S., Stein, W. and Weigandt, J., http://wstein.org/papers/2016-height/.
2. Bektemirov, B., Mazur, B., Stein, W. and Watkins, M., ‘Average ranks of elliptic curves: tension between data and conjecture’, Bull. Amer. Math. Soc. (N.S.) 44 (2007) no. 2, 233254.
3. Bennett, M. and Rechnitzer, A., ‘Computing elliptic curves over $\mathbb{Q}$ : bad reduction at one prime’, Preprint.
4. Bhargava, M. and Ho, W., ‘On the average sizes of Selmer groups in families of elliptic curves’, Preprint.
5. Bhargava, M., Kane, D. M., Lenstra, H. W. Jr., Poonen, B. and Rains, E., ‘Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves’, Camb. J. Math. 3 (2015) no. 3, 275321.
6. Bhargava, M. and Shankar, A., ‘Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves’, Ann. of Math. (2) 181 (2015) no. 1, 191242.
7. Bhargava, M. and Shankar, A., ‘Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0’, Ann. of Math. (2) 181 (2015) no. 2, 587621.
8. Bhargava, M. and Shankar, A., ‘The average number of elements in the $4$ -Selmer groups of elliptic curves is $7$ ’, Preprint, 2013, arXiv:1312.7333.
9. Bhargava, M. and Shankar, A., ‘The average size of the $5$ -Selmer group of elliptic curves is $6$ , and the average rank is less than $1$ ’, Preprint, 2013, arXiv:1312.7859.
10. Birch, B. J. and Kuyk, W. (eds), Modular functions of one variable, Vol. IV , Lecture Notes in Mathematics 476 (Springer, Berlin–New York, 1975).
11. Bober, J. W., ‘Conditionally bounding analytic ranks of elliptic curves’, ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium , Open Book Series 1 (Mathematical Sciences Publishers, Berkeley, CA, 2013) 135144.
12. Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235265, Computational algebra and number theory (London, 1993).
13. Brumer, A., ‘The average rank of elliptic curves. I’, Invent. Math. 109 (1992) no. 3, 445472.
14. Brumer, A. and McGuinness, O., ‘The behavior of the Mordell-Weil group of elliptic curves’, Bull. Amer. Math. Soc. (N.S.) 23 (1990) no. 2, 375382.
15. Cremona, J., Algorithms for modular elliptic curves , 2nd edn (Cambridge University Press, Cambridge, 1997).
16. Cremona, J., ecdata: 2016-02-07, http://dx.doi.org/10.5281/zenodo.45657, February 2016.
17. Delaunay, C., ‘Heuristics on Tate-Shafarevitch groups of elliptic curves defined over ℚ’, Experiment. Math. 10 (2001) no. 2, 191196.
18. Delaunay, C. and Jouhet, F., ‘ p -torsion points in finite abelian groups and combinatorial identities’, Adv. Math. 258 (2014) 1345.
19. Goldfeld, D., ‘Conjectures on elliptic curves over quadratic fields’, Number theory, Carbondale , Proceedings of Southern Illinois Conference, Southern Illinois University, Carbondale, Ill, 1979, Lecture Notes in Mathematics 751 (Springer, Berlin, 1979) 108118.
20. Harron, R. and Snowden, A., ‘Counting elliptic curves with prescribed torsion’, J. reine angew. Math., to appear, doi:10.1515/crelle-2014-0107.
21. Heath-Brown, D. R., ‘The average analytic rank of elliptic curves’, Duke Math. J. 122 (2004) no. 3, 591623, 04.
22. Iwaniec, H. and Kowalski, E., Analytic number theory , American Mathematical Society Colloquium Publications 53 (American Mathematical Society, Providence, RI, 2004).
23. Katz, N. M. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy , American Mathematical Society Colloquium Publications 45 (American Mathematical Society, Providence, RI, 1999).
24. Mestre, J.-F., ‘Formules explicites et minorations de conducteurs de variétés algébriques’, Compositio Math. 58 (1986) no. 2, 209232.
25. Park, J., Poonen, B., Voight, J. and Wood, M., ‘Heuristics for boundedness of ranks of elliptic curves over $\mathbb{Q}$ ’, Preprint, 2016, arXiv:1602.01431.
26. Poonen, B. and Rains, E., ‘Random maximal isotropic subspaces and Selmer groups’, J. Amer. Math. Soc. 25 (2012) no. 1, 245269.
27. SageMath, Inc. SageMathCloud, 2016. https://cloud.sagemath.com.
28. Spicer, S. V., The zeros of elliptic curve $L$ -functions: analytic algorithms with explicit time complexity, PhD Thesis, University of Washington, 2015.
29. Stein, W. A. and Watkins, M., ‘A database of elliptic curves—first report’, Algorithmic number theory (Sydney, 2002) , Lecture Notes in Computer Science 2369 (Springer, Berlin, 2002) 267275.
30. The SageMath Developers. Sage Mathematics Software (Version 6.9), 2015. http://www.sagemath.org.
31. Watkins, M., ‘Some heuristics about elliptic curves’, Experiment. Math. 17 (2008) no. 1, 105125.
32. Young, M., ‘Low-lying zeros of families of elliptic curves’, J. Amer. Math. Soc. 19 (2006) no. 1, 205250.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

  • Jennifer S. Balakrishnan (a1), Wei Ho (a2), Nathan Kaplan (a3), Simon Spicer (a4), William Stein (a5) and James Weigandt (a6)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed