Tame Galois realizations of \(\text{GL}_2(\mathbb F_{\ell})\) over \(\mathbb Q\)
type: Item
instance of: scholarly article
title: Tame Galois realizations of \(\text{GL}_2(\mathbb F_{\ell})\) over \(\mathbb Q\)
zbMATH Open document ID: 1193.11106
author: Sara Arias-de-Reyna
author: Nuria Vila
published in: Journal of Number Theory
publication date: 2009-05-19T00:00:00Z
review text: The present paper deals with the following stronger version, proposed by B. Birch, of the Inverse Galois Problem over \(\mathbb Q\): given a finite group \(G\), is there a tamely ramified Galois extension \(K/\mathbb Q\) with \(\text{Gal}(K/\mathbb Q)\cong G\)? The authors prove that the answer to this question is `yes' for the group \(G=\text{GL}_2(\mathbb F_\ell)\), for every prime number \(\ell\). Firstly, they show that a tame realization of \(G=\text{GL}_2(\mathbb F_\ell)\) arises from the Galois representation \(\rho_\ell: \text{Gal}(\overline{\mathbb Q}/\mathbb Q) \rightarrow \Aut(E[\ell])\) attached to the \(\ell\)-torsion of an elliptic curve \(E\) over \(\mathbb Q\) provided that: 1) \(E\) is semistable; 2) \(E\) has good supersingular reduction at \(\ell\); 3) \(\rho_\ell\) is surjective. They then prove that, for each prime number \(\ell\), there exist infinitely many elliptic curves over \(\mathbb Q\) such that 1), 2) and 3) hold. For \(\ell <11\), this follows from the joint work of \textit{A. Reverter} and the second author [Can. Math. Bull. 44, No. 3, 313--322 (2001; Zbl 1028.11035)]. For \(\ell \geq 11\), property 3) follows from property 1) by a result of B. Mazur, and the authors show how to construct (infinitely many) elliptic curves over \(\mathbb Q\) satisfying 1) (hence 3)) and 2). In addition, they illustrate their method with numerical examples for \(\ell \in \{11,13,17\}\).
reviewed by: Bernat Plans
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 5554695
zbMATH Keywords: tame ramification
zbMATH Keywords: Galois extension
zbMATH Keywords: supersingular elliptic curve
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.09.020
OpenAlex ID: W2067312586
cites work: Legendre elliptic curves over finite fields
cites work: Q4325460
cites work: Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial
cites work: Congruence properties of special elliptic functions
cites work: Q4003857
cites work: The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb Q\).
cites work: Counting nilpotent Galois extensions
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: Q4808264
cites work: Tame \(A_{n}\)-extensions of \({\mathbb Q}\)
cites work: Images of \(\text{mod }p\) Galois representations associated to elliptic curves
cites work: Q5527558
cites work: Galois properties of points of finite order of elliptic curves
cites work: Q5286654
cites work: Q3997322
cites work: Q3710637
DOI: 10.1016/J.JNT.2008.09.020
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description: scientific article; zbMATH DE number 5554695