Remarks on mod-\(l^n\) representations, \(l=3,5\)
type: Item
instance of: scholarly article
title: Remarks on mod-\(l^n\) representations, \(l=3,5\)
zbMATH Open document ID: 0954.11020
author: Siman Wong
author: Brian Conrad
published in: Journal of Number Theory
publication date: 2001-02-12T00:00:00Z
review text: Let \(\ell\) be an odd prime and \(n > 1\) an integer. For an elliptic curve \(E/K\) over a number field \(K\), denote by \(a_{\mathfrak p}(E)\) the trace on the Tate module \(T_\ell(E)\) of an arithmetic Frobenius element at \(\mathfrak p\). Suppose that the associated mod-\(\ell^n\) Galois representation \(\rho_{E, \ell^n} : G_K \to \mathop{GL}_2({\mathbb Z}/\ell^n {\mathbb Z})\) is surjective. In an earlier paper by \textit{S. Wong} [J. Number Theory 74, 1-18 (1999; Zbl 0931.11016)], it was proved that for \(\ell > 5\), \(\rho_{E, \ell^n}\) is determined up to twisting by a continuous character by the set of primes \(\mathfrak p\) outside a finite set such that \(\ell^n\) divides \(a_{\mathfrak p}\). In the paper under review, the authors complement this result by showing that it is no longer true for \(\ell = 3\) and \(\ell = 5\). More precisely, there is a number field \(L\) of absolute degree \(\ell^{3(n-1)}\) such that there are infinitely many pairs \((E_1, E_2)\) of elliptic curves over \(L\), all with distinct \(j\)-invariants, such that (1) \(\rho_{E_1, \ell^n}\) and \(\rho_{E_2, \ell^n}\) are both surjective, (2) for almost all primes \(\mathfrak p\) of \(L\), \(\ell^n\) divides \(a_{\mathfrak p}(E_1)\) if and only if it divides \(a_{\mathfrak p}(E_2)\), (3) \(\rho_{E_1, \ell^n}\) and \(\rho_{E_2, \ell^n}\) are not related by twisting by a continuous character \(G_L \to ({\mathbb Z}/\ell^n {\mathbb Z})^\times\). For the proof, one starts with a curve \(E/K\) and its representation \(\rho = \rho_{E, \ell^n}\). The presence of a non-trivial outer automorphism of \(\mathop{PGL}_2({\mathbb Z}/\ell^n {\mathbb Z})\) allows to construct another representation \(\rho'\) such that \(\rho\) and \(\rho'\) have the required properties. The final step is to realize \(\rho'\) as a representation coming from an elliptic curve. This is done by analyzing the corresponding modular curve \(X(\rho')\). At this point, it is necessary to enlarge the base field in order to obt…
reviewed by: Michael Stoll
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1375090
zbMATH Keywords: Galois representations
zbMATH Keywords: elliptic curves
zbMATH Keywords: twist by character
MaRDI profile type: Publication
full work available at URL: jnth.1999.2406
OpenAlex ID: W2118797966
cites work: Automorphisms of the Two-Dimensional Linear Groups Over Integral Domains
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Troisième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Arithmetic Moduli of Elliptic Curves. (AM-108)
cites work: Q3309989
cites work: Q4317891
cites work: Twists of Galois representations and projective automorphisms
DOI: 10.1006/JNTH.1999.2406
description: scientific article; zbMATH DE number 1375090