Tame Galois realizations of \(\text{GL}_2(\mathbb F_{\ell})\) over \(\mathbb Q\)
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 \(\ell \in \{11,13,17\}\).
scientific article; zbMATH DE number 5554695
scientific article