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Reductions of an elliptic curve and their Tate-Shafarevich groups

โœ Scribed by A. C. Cojocaru; W. Duke


Book ID
105872973
Publisher
Springer
Year
2004
Tongue
English
Weight
207 KB
Volume
329
Category
Article
ISSN
0025-5831

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Let E be an elliptic curve over ] be an irreducible polynomial of odd degree, and let K =F ( โˆš d). Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(EโŠ— F K, 1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group I(E/K

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We compute the ,-Selmer group for a family of elliptic curves, where , is an isogeny of degree 5, then find a practical formula for the Cassels Tate pairing on the ,-Selmer groups and use it to show that a particular family of elliptic curves have non-trivial 5-torsion in their Shafarevich Tate grou