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ON THE TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER ?

  • Kim, Do-Hyeong (Department of Mathematics Pohang University of Science and Technology)
  • Received : 2010.09.24
  • Published : 2012.01.31

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

References

  1. G. Chinta, Analytic ranks of elliptic curves over cyclotomic fields, J. Reine Angew. Math. 544 (2002), 13-24.
  2. 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. https://doi.org/10.1007/s00032-010-0127-2
  3. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford Science Publications, 1938.
  4. K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Asterisque 295 (2004), 117-290.
  5. J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19-66.
  6. 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. https://doi.org/10.1007/BF01388731
  7. J. Silverman, The Arithmetic of Elliptic Curves, 2nd edition, Graduate Texts in Mathematics, vol. 106, Springer, 2008.

Cited by

  1. ON THE p-PRIMARY PART OF TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER ℚ WHEN p IS SUPERSINGULAR vol.50, pp.2, 2013, https://doi.org/10.4134/BKMS.2013.50.2.407