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


Author: Gang Yu
Journal: Trans. Amer. Math. Soc. 358 (2006), 1563-1584
MSC (2000): Primary 11G05, 14H52
DOI: https://doi.org/10.1090/S0002-9947-05-03806-7
Published electronically: October 31, 2005
MathSciNet review: 2186986
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. R. Bölling, Die Ordnung der Schafarewitsch-Tate Gruppe kann beliebig gross werden, Math. Nachr. 67 (1975), 157-179. MR 0384812 (52:5684)
  • 2. A. Brumer, The average rank of elliptic curves, I, Invent. Math 109 (1992), 445-472. MR 1176198 (93g:11057)
  • 3. D.A.Burgess, On character sums and $ L$-series, II, Proc. Lond. Math. Soc., III. Ser.13, (1963), 524-536. MR 0148626 (26:6133)
  • 4. H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, 1974. MR 0424730 (54:12689)
  • 5. D. R. Heath-Brown, The size of Selmer groups for the congruent number problem I, Invent. Math. 111 (1) (1993), 171-195. MR 1193603 (93j:11038)
  • 6. -, The size of Selmer groups for the congruent number problem II, Invent. Math. 118 (2) (1994), 331-370. MR 1292115 (95h:11064)
  • 7. A. Ivic, The Riemann zeta-function, John Wiley & Sons, 1985. MR 0792089 (87d:11062)
  • 8. K. Kramer, A family of semistable elliptic curves with large Tate-Shafarevitch groups, Proc. Amer. Math Soc. 89 (1983), 379-386. MR 0715850 (85d:14059)
  • 9. A. Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. 18 (1954), 83-87. MR 0067143 (16:676a)
  • 10. J. Silverman, The arithmetic of elliptic curves, GTM 106, Springer, 1986. MR 0817210 (87g:11070)
  • 11. G. Yu, Rank 0 quadratic twists of a family of elliptic curves, Compositio Math 135 (3) (2003), 331-356. MR 1956817 (2004b:11082)
  • 12. -, Average size of $ 2$-Selmer groups of elliptic curves, II, to appear in Acta Arith.

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Additional Information

Gang Yu
Affiliation: Department of Mathematics, LeConte College, 1523 Greene Street, University of South Carolina, Columbia, South Carolina 29208
Email: yu@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03806-7
Keywords: Elliptic curves, $2$-descent procedure, character sums
Received by editor(s): September 16, 2000
Received by editor(s) in revised form: May 2, 2004
Published electronically: October 31, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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