On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

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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⊗F K, 1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group (E/K) when L (E⊗FK, 1) ≠ 0.

Original languageEnglish (US)
Pages (from-to)249-283
Number of pages35
JournalJournal of Number Theory
Issue number2
Publication statusPublished - Dec 1 2005


All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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