[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2012.49.1.155

ON THE TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER ?

Kim, Do-Hyeong (Department of Mathematics Pohang University of Science and Technology)

Publication Information

Bulletin of the Korean Mathematical Society / v.49, no.1, 2012 , pp. 155-163 More about this Journal

Abstract

Let E be an elliptic curve over $\mathbb{Q}$ . Using Iwasawa theory, we give what seems to be the first general upper bound for the order of vanishing of the p-adic L-function at s = 0, and the $\mathbb{Z}_p$ -corank of the Tate-Shafarevich group for all sufficiently large good ordinary primes p.

Keywords

good ordinary reduction; Tate-Shafarevich group; elliptic curves; Iwasawa theory;

Citations & Related Records

(Related Records In Web of Science)
Times Cited By SCOPUS : 0

Reference

1	J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19-66.
2	B. Mazur, J. Tate, and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1-48. DOI ScienceOn
3	J. Silverman, The Arithmetic of Elliptic Curves, 2nd edition, Graduate Texts in Mathematics, vol. 106, Springer, 2008.
4	G. Chinta, Analytic ranks of elliptic curves over cyclotomic fields, J. Reine Angew. Math. 544 (2002), 13-24.
5	J. Coates, Z. Liang, and R. Sujatha, The Tate-Shafarevich group for elliptic curves with complex multiplication II, Milan J. Math. 78 (2010), no. 2, 395-416. DOI ScienceOn
6	G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford Science Publications, 1938.
7	K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque 295 (2004), 117-290.