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

Mihran Papikian

doi:10.1016/j.jnt.2004.11.006

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

Mihran Papikian

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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⊗_FK, 1) ≠ 0.

Original language	English (US)
Pages (from-to)	249-283
Number of pages	35
Journal	Journal of Number Theory
Volume	115
Issue number	2
DOIs	https://doi.org/10.1016/j.jnt.2004.11.006
State	Published - Dec 2005

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Access to Document

10.1016/j.jnt.2004.11.006

Cite this

@article{ceb86d48d7fb4cfebba6df00aa372f89,

title = "On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves",

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.",

author = "Mihran Papikian",

note = "Funding Information: Supported in part by the European Postdoctoral Institute Fellowship. E-mail address: [email protected].",

year = "2005",

month = dec,

doi = "10.1016/j.jnt.2004.11.006",

language = "English (US)",

volume = "115",

pages = "249--283",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

number = "2",

}

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: [email protected].

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.

UR - https://www.scopus.com/pages/publications/28444471229

UR - https://www.scopus.com/pages/publications/28444471229#tab=citedBy

U2 - 10.1016/j.jnt.2004.11.006

DO - 10.1016/j.jnt.2004.11.006

M3 - Article

AN - SCOPUS:28444471229

SN - 0022-314X

VL - 115

SP - 249

EP - 283

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 2

ER -

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

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this