Average size of 2-Selmer groups of elliptic curves, I

Yu, Gang

doi:10.1090/S0002-9947-05-03806-7

Average size of $2$-Selmer groups of elliptic curves, I
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Trans. Amer. Math. Soc. 358 (2006), 1563-1584 Request permission

Abstract:

In this paper, we study a class of elliptic curves over $\mathbb {Q}$ with $\mathbb {Q}$-torsion group ${\mathbb {Z}}_{2}\times \mathbb {Z}_{2}$, and prove that the average order of the $2$-Selmer groups is bounded.

References

Reinhard Bölling, Die Ordnung der Schafarewitsch-Tate-Gruppe kann beliebig großwerden, Math. Nachr. 67 (1975), 157–179 (German). MR 384812, DOI 10.1002/mana.19750671003
Armand Brumer, The average rank of elliptic curves. I, Invent. Math. 109 (1992), no. 3, 445–472. MR 1176198, DOI 10.1007/BF01232033
D. A. Burgess, On character sums and $L$-series. II, Proc. London Math. Soc. (3) 13 (1963), 524–536. MR 148626, DOI 10.1112/plms/s3-13.1.524
H. Halberstam and H.-E. Richert, Sieve methods, London Mathematical Society Monographs, No. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1974. MR 0424730
D. R. Heath-Brown, The size of Selmer groups for the congruent number problem, Invent. Math. 111 (1993), no. 1, 171–195. MR 1193603, DOI 10.1007/BF01231285
D. R. Heath-Brown, The size of Selmer groups for the congruent number problem. II, Invent. Math. 118 (1994), no. 2, 331–370. With an appendix by P. Monsky. MR 1292115, DOI 10.1007/BF01231536
Aleksandar Ivić, The Riemann zeta-function, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the Riemann zeta-function with applications. MR 792089
Kenneth Kramer, A family of semistable elliptic curves with large Tate-Shafarevitch groups, Proc. Amer. Math. Soc. 89 (1983), no. 3, 379–386. MR 715850, DOI 10.1090/S0002-9939-1983-0715850-1
Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. (N.S.) 18 (1954), 83–87. MR 67143
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Gang Yu, Rank 0 quadratic twists of a family of elliptic curves, Compositio Math. 135 (2003), no. 3, 331–356. MR 1956817, DOI 10.1023/A:1022258905572
—, Average size of $2$-Selmer groups of elliptic curves, II, to appear in Acta Arith.