Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Gang Yu
Journal: Trans. Amer. Math. Soc. 358 (2006), 1563-1584.
MSC (2000): Primary 11G05, 14H52
Posted: October 31, 2005
Retrieve article in: PDF DVI PostScript

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:

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.

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G05, 14H52

Retrieve articles in all Journals with MSC (2000): 11G05, 14H52

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: 10.1090/S0002-9947-05-03806-7
PII: S 0002-9947(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
Posted: October 31, 2005
Copyright of article: Copyright 2005, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google