Twisted root numbers of elliptic curves semistable at primes above 2 and 3
type: Item
instance of: scholarly article
title: Twisted root numbers of elliptic curves semistable at primes above 2 and 3
zbMATH Open document ID: 1205.11064
author: Ryota Matsuura
published in: Algebra \& Number Theory
publication date: 2010-05-06T00:00:00Z
review text: Let \(E\) be an elliptic curve defined over a number field \(F\). Fix a prime number \(p\) and consider the field \(F^\infty = F(E[p^\infty])\) generated by the \(p\)-power torsion points. Let \(\tau\) be an irreducible self-dual representation of Gal(\(F^\infty/F\)), let \(L(s,E,\tau)\) be the associated L-function and \(W(E,\tau) = (-1)^w\) the corresponding root number. The main result of this article based on the author's dissertation gives explicit formulas for \(w\) modulo \(2\) which are similar to (but more general than) the formulas obtained by \textit{D. E. Rohrlich} [Math. Res. Lett. 13, No. 2--3, 359--376 (2006; Zbl 1124.11026)]. As an application of his results, the author proves that if \(E\) is an elliptic curve over the rationals, semistable at \(2\) and \(3\), and if \(p \equiv 3 \bmod 4\) is a sufficiently large prime, then the rank of \(E\) is unbounded in its \(p\)-division tower.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 5704470
zbMATH Keywords: elliptic curve
zbMATH Keywords: torsion points
zbMATH Keywords: L-function
zbMATH Keywords: root number
zbMATH Keywords: Galois representation
zbMATH Keywords: Mordell-Weil rank
MaRDI profile type: Publication
full work available at URL: ant.2010.4.255
OpenAlex ID: W1978012678
DOI: 10.2140/ANT.2010.4.255
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description: scientific article; zbMATH DE number 5704470