Root numbers of semistable elliptic curves in division towers
type: Item
instance of: scholarly article
title: Root numbers of semistable elliptic curves in division towers
zbMATH Open document ID: 1124.11026
author: David E. Rohrlich
published in: Mathematical Research Letters
publication date: 2007-02-15T00:00:00Z
review text: Let \(F\) be a number field. Let \(E/F\) an elliptic curve. Denote \(F^\infty=F(E[p^\infty])\) the field of all \(p\)-division points on \(E\). The paper under review discusses the root number \(W(E,\tau)\) of the \(L\)-function \(L(s,E,\tau)\) where \(\tau\) is an irreducible self-dual representation of \(F^\infty/F\). Under some assumptions \(E/F\) is semistable, \(p\) is odd, the natural map \(\text{Gal}(F^\infty/F) \hookrightarrow \text{GL}(T_p)\cong \text{GL}_2(\mathbb Z_p)\) is an isomorphism, and for a bad place \(v\) we have that \(\text{ord}_v j(E) \not \equiv 0 \bmod p\)) the root number \(W(E,\tau)\) is computed in terms of invariants of \(E\) and \(F\). The proof consists of a detailed study of linear representations of \(\text{GL}(2,\mathbb Z_p)\).
reviewed by: Remke Kloosterman
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 5127173
zbMATH Keywords: Elliptic curves
zbMATH Keywords: root number
zbMATH Keywords: Mordell-Weil rank
MaRDI profile type: Publication
full work available at URL: mrl.2006.v13.n3.a3
OpenAlex ID: W2042397071
DOI: 10.4310/MRL.2006.V13.N3.A3
description: scientific article; zbMATH DE number 5127173