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

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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
Volume115
Issue number2
DOIs
StatePublished - Dec 1 2005

Fingerprint

Hyperelliptic Curves
Elliptic Curves
Irreducible polynomial
Prime Ideal
Conductor
Gross
Odd
Analogue

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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abstract = "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.",
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On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves. / Papikian, Mihran.

In: Journal of Number Theory, Vol. 115, No. 2, 01.12.2005, p. 249-283.

Research output: Contribution to journalArticle

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