Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves |
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Authors: | Byungchul Cha |
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Institution: | Department of Mathematics and Computer Science, Hendrix College, 1600 Washington Ave, Conway, AR 72032, USA |
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Abstract: | Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H1(Gal(K(E?i])/K),E?i]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture. |
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Keywords: | Galois cohomology Elliptic curves Birch and Swinnerton-Dyer conjecture Shafarevich-Tate groups |
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