On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve
type: Item
instance of: scholarly article
title: On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve
zbMATH Open document ID: 0661.14033
DOI: 10.1007/BF01410203
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143612
review text: Let \(F_{a,b,c}\) be a complete non-singular model over \({\mathbb{Q}}\) of the plane curve defined by the equation \(y^ p=x^ a(1-x)^ b\), where p is an odd prime integer and a,b,c are integers with \(0<a,b<p\) and \(a+b+c=0\). The author shows that the Shafarevich-Tate group {Russian{Sh}} of the Jacobian variety over \({\mathbb{Q}}(\mu_ p)\) of \(F_{a,b,c}\) contains a subgroup isomorphic to (\({\mathbb{Z}}/p{\mathbb{Z}})\oplus ({\mathbb{Z}}/p{\mathbb{Z}})\), provided that \(p\equiv 1\quad (\pmod 4),\) \(p\nmid B_{(p-1)/2}\cdot B_{(p+3)/2}\) and \(-2abc\{(a^ ab^ bc^ c)^{p-1}-1\}/p\in {\mathbb{F}}_ p^{*2}\), where \(B_ i\) is the i-th Bernoulli number. He gets the result by showing that the Cassels-Tate pairing is non-trivial in the case of \(F_{a,b,c}\) over \({\mathbb{Q}}_ p(\mu_ p)\) having a wild split reduction.
Mathematics Subject Classification ID: 14K30
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 4079614
zbMATH Keywords: Fermat curve
zbMATH Keywords: Shafarevich-Tate group
zbMATH Keywords: Cassels-Tate pairing
author: William G. McCallum
reviewed by: Tsutomu Sekiguchi
MaRDI profile type: Publication
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OpenAlex ID: W1999604759
description: scientific article; zbMATH DE number 4079614