Realization of some Galois representations of low degree in Mordell-Weil groups
type: Item
instance of: scholarly article
title: Realization of some Galois representations of low degree in Mordell-Weil groups
zbMATH Open document ID: 0881.11052
author: David E. Rohrlich
published in: Mathematical Research Letters
publication date: 1998-01-06T00:00:00Z
review text: Let \(\rho\) be a \(\mathbb{Q}\)-representation of a group \(G\). Suppose \(K/M\) is a Galois extension of number fields such that \(\text{Gal}(K/M)\cong G\), so that we may view \(\rho\) as a representation of \(\text{Gal}(K/M)\). If \(E/M\) is an elliptic curve, we will say that \(\rho\) occurs in \(E(K)\) if \(\rho\) is isomorphic to a subrepresentation of the natural representation of \(\text{Gal}(K/M)\) on \(E(K)\otimes_{\mathbb{Z}}{\mathbb{Q}}\). In the paper under review, the author shows that for certain \(G\) and \(\rho\), one may choose an arbitrary extension \(K/M\) with \(\text{Gal}(K/M)\cong G\) and be guaranteed that there exists an \(E/M\) such that \(\rho\) occurs in \(E(K)\). In particular, the author's main theorem can be stated as follows: Suppose \(G\) is a group with a subgroup \(H\) of index at most \(9\), and let \(\rho\) be the \(\mathbb{Q}\)-representation of \(G\) determined (up to isomorphism) by \[ \text{ind}_H^G\text{\textbf{1}}_H\cong\text{\textbf{1}}_G\oplus\rho, \] where \(\text{\textbf{1}}_H\) and \(\text{\textbf{1}}_G\) are the trivial representations of \(H\) and \(G\) and where \(\text{ind}_H^G\) is the induction functor from representations of \(H\) to representations of \(G\). Let \(K/M\) be an extension of number fields with \(\text{Gal}(K/M)\cong G\). Then there exists an elliptic curve \(E/M\) such that \(\rho\) occurs in \(E(K)\). The proof is by a clever direct construction. Let \(L\) be the subfield of \(K\) corresponding to the subgroup \(H\) of \(G\). One picks an element \(\xi\) of \(L\) that satisfies local conditions at two auxiliary primes \(p\) and \(q\), and uses this \(\xi\) to write down an elliptic curve \(E\) over \(M\) and a certain point \(P\) in \(E(L)\). The conditions at \(p\) and \(q\) are then used to show that the sub-\(G\)-module of \(E(K)\otimes_{\mathbb{Z}}{\mathbb{Q}}\) generated by \(P\) gives the desired representation. \textit{T. Shioda} [Mordell-Weil lattices and Galois representation I, II, III…
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 1019791
zbMATH Keywords: Mordell-Weil lattice
zbMATH Keywords: Galois representation
zbMATH Keywords: elliptic curve
reviewed by: Everett W. Howe
MaRDI profile type: Publication
DOI: 10.4310/MRL.1997.V4.N1.A11
description: scientific article; zbMATH DE number 1019791