Twists of Galois representations and projective automorphisms
type: Item
instance of: scholarly article
title: Twists of Galois representations and projective automorphisms
zbMATH Open document ID: 0931.11016
author: Siman Wong
published in: Journal of Number Theory
publication date: 2000-02-22T00:00:00Z
review text: Let \(\mathcal O\) be a commutative, complete local ring with residue field \(k\) and maximal ideal \(\lambda\). Also, let \(K\) be a finite extension of the rationals \(\mathbb Q\). Write \(G_K:=\text{Gal}(\overline{K}/K)\) and let \(\rho_1,\rho_2:G_K\rightarrow GL_n(\mathcal O)\) be surjective, continuous Galois representations, unramified outside a finite set of primes \(S\). Let \(a_i(p):=\text{Tr}(\rho_i(\text{Frob}_p))\), \(i=1,2\), \(p\not\in S\). The \(\rho_i\) are said to be \(\lambda\)-adically close near the supersingular primes (CSP, for short) if there exists an integer \(N_0>0\) such that for any integer \(w>0\), and primes \(p\) of \(K\) not in \(S\), if both \(a_i(p)\in\lambda^{N_0}\), then \(\lambda^w|a_1(p)\iff\lambda^w|a_2(p)\). One likes to know whether two such representations are `the same', i.e. are equal up to the following properties: (i) twist by a continuous character \(G_K\rightarrow{\mathcal O}^{\times}\); (ii) conjugation by a matrix in \(GL_n(\mathcal O)\); (iii) the transpose inverse automorphism of \(GL_n(\mathcal O)\); and (iv) a ring automorphism of \(\mathcal O\). With the above terminology, one can state the main result \((*)\): \(\rho_1\) and \(\rho_2\) are the same up to (i), (ii), (iii) or (iv) whenever: \((1)\) \(\mathcal O\) is a domain and \(k\not\simeq{\mathbb F}_2,{\mathbb F}_3\), or, \((2)\) \(\mathcal O\) is noetherian, \(n\geq 4\) is even and \(\text{char}(k)>2\); or \(n=2\) and \(k\not\simeq{\mathbb F}_2,{\mathbb F}_3,{\mathbb F}_5\), or \((3)\) \(\mathcal O\) is noetherian, \(n\) is odd, \(k\not\simeq{\mathbb F}_2\), and, if \(n=3\), \(k\not\simeq{\mathbb F}_3\). In the case \(n=3\) and \(k\simeq{\mathbb F}_3\) or \(n=2\) and \(\mathcal O={\mathbb Z}/\ell^N{\mathbb Z}\), \(\ell=3\) or \(5\), the result holds for the pair of representations \(G_K\rightarrow GL_n({\mathbb Z}/\ell^{N-1}{\mathbb Z})\) induced from the \(\rho_i\). The proof relies on the knowledge of the automorphisms of \(PGL_n(\mathcal O)\). This is the…
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1271761
zbMATH Keywords: Galois representation
zbMATH Keywords: twist
zbMATH Keywords: elliptic curve
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
full work available at URL: jnth.1998.2307
OpenAlex ID: W2029061337
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DOI: 10.1006/JNTH.1998.2307
description: scientific article; zbMATH DE number 1271761