Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T00:58:41.250Z Has data issue: false hasContentIssue false
  • English
  • Français

On the φ-Selmer groups of the elliptic curves y2 = x3 − Dx

Published online by Cambridge University Press:  09 September 2016

DANIEL M. KANE  and
JACK A. THORNE
DANIEL M. KANE
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0404, La Jolla, CA 92093-0404. e-mail: [email protected]
JACK A. THORNE
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, CB3 0WB, U.K. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the variation of the φ-Selmer groups of the elliptic curves y2 = x3 − Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when the integer D is constrained to lie in a family for which the relative Tamagawa number of the isogeny φ is fixed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 71 - 93
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bhargava, M. and Shankar, A.. Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Ann. Math. 181 (1) (2015), 191–242.CrossRefGoogle Scholar
[2] Birch, B. J. and Stephens, N. M.. The parity of the rank of the Mordell–Weil group. Topology 5 (1966), 295–299.Google Scholar
[3] Cassels, J. W. S. Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton–Dyer. J. Reine Angew. Math. 217 (1965), 180–199.Google Scholar
[4] Goto, T. A note on the Selmer group of the elliptic curve y 2 = x 3 + Dx . Proc. Japan Acad. Ser. A Math. Sci. 77 (7) (2001), 122–125.Google Scholar
[5] Hardy, G. and Ramanujan, S.. The normal number of prime factors of a number. Quarterly J. Math. 48 (1917), 76–92.Google Scholar
[6] Jacobi, K. G. J. Fundamenta Nova Theoriae Functionum Ellipticarum (Königsberg, 1829).Google Scholar
[7] Kane, D. M. On the ranks of the 2-Selmer groups of twists of a given elliptic curve. Algebra Number Theory 7 (5) (2013), 1253–1279.Google Scholar
[8] Karlin, S. and McGregor, J.. Random walks. Illinois J. Math. 3 (1959), 66–81.Google Scholar
[9] Norris, J. R. Markov chains. Cambridge Series in Statistical and Probabilistic Mathematics vol. 2 (Cambridge University Press, Cambridge, 1998). Reprint of 1997 original.Google Scholar
[10] Poonen, B. and Rains, E.. Random maximal isotropic subspaces and Selmer groups. J. Amer. Math. Soc. 25 (1) (2012), 245–269.Google Scholar
[11] Silverman, J. H. The arithmetic of elliptic curves. Graduate Texts in Math. vol. 106 (Springer, Dordrecht, second edition, 2009).Google Scholar
[12] Swinnerton-Dyer, P. The effect of twisting on the 2-Selmer group. Math. Proc. Camb. Phil. Soc. 145 (3) (2008), 513–526.Google Scholar
[13] van Doorn, E. A. and Schrijner, P.. Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes. J. Austral. Math. Soc. Ser. B 37 (2) (1995), 121–144.Google Scholar