TY - JOUR

T1 - On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

AU - Papikian, Mihran

N1 - Funding Information:
Supported in part by the European Postdoctoral Institute Fellowship. E-mail address: papikian@math.stanford.edu.

PY - 2005/12

Y1 - 2005/12

N2 - 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.

AB - 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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U2 - 10.1016/j.jnt.2004.11.006

DO - 10.1016/j.jnt.2004.11.006

M3 - Article

AN - SCOPUS:28444471229

VL - 115

SP - 249

EP - 283

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -