ON THE p-PRIMARY PART OF TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER Q WHEN p IS SUPERSINGULAR
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Let E be an elliptic curve over Q and p be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic Zp-extension of Q is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the number of copies of Qp/Zp occurring in the p-primary part of the Tate-Shafarevich group of E over Q.
Abstract
Let E be an elliptic curve over Q and p be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic Zp-extension of Q is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the number of copies of Qp/Zp occurring in the p-primary part of the Tate-Shafarevich group of E over Q.
참고문헌 (8)
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- J. Coates, "The Tate-Shafarevich group for elliptic curves with complex multiplication II", Milan J. Math, 78(2), pp.395~416 (2010)
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- 김도형, "ON THE TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER Q", 대한수학회보, 49(1), pp.155~163 (2012), 대한수학회
- S. Kobayashi, "Iwasawa theory for elliptic curves at supersingular primes", Invent. Math, 152(1), pp.1~36 (2003)
- B. Mazur, "Arithmetic of Weil Cuves", Invent. Math, 25, pp.1~61 (1974)
- B. Mazur, "On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer", Invent. Math, 84(1), pp.1~48 (1986)
- R. Pollack, "On the p-adic L-function of a modular form at a supersingular prime", Duke Math. J, 118(3), pp.523~558 (2003)