On the variation of Tate–Shafarevich groups of elliptic curves over hyperelliptic curves
نویسنده
چکیده
Let E be an elliptic curve over F=Fq(t) having conductor (p)·∞, where (p) is a prime ideal in Fq [t]. Let d ∈ Fq [t] be an irreducible polynomial of odd degree, and let K=F( √ d). Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗FK, 1). As a consequence, we obtain a formula for the order of the Tate–Shafarevich group I(E/K) when L(E⊗FK, 1) = 0. © 2005 Elsevier Inc. All rights reserved. MSC: primary 11G05, 11G40; secondary 11G18
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تاریخ انتشار 2005