### Abstract

Let E be an elliptic curve over F = F_{q}(t) having conductor (p)·∞, where (p) is a prime ideal in F_{q}[t]. Let d ∈ F_{q}[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⊗_{F}K, 1) ≠ 0.

Original language | English (US) |
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Pages (from-to) | 249-283 |

Number of pages | 35 |

Journal | Journal of Number Theory |

Volume | 115 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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*Journal of Number Theory*, vol. 115, no. 2, pp. 249-283. https://doi.org/10.1016/j.jnt.2004.11.006

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

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Papikian, Mihran

PY - 2005/12/1

Y1 - 2005/12/1

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.

UR - http://www.scopus.com/inward/record.url?scp=28444471229&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28444471229&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2004.11.006

DO - 10.1016/j.jnt.2004.11.006

M3 - Article

VL - 115

SP - 249

EP - 283

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -