Tame Galois realizations of \(\text{GL}_2(\mathbb F_{\ell})\) over \(\mathbb Q\). (English) Zbl 1193.11106
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 \(\rho_\ell: \text{Gal}(\overline{\mathbb Q}/\mathbb Q) \rightarrow \operatorname{Aut}(E[\ell])\) 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 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 \(\ell \in \{11,13,17\}\).
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 \(\rho_\ell: \text{Gal}(\overline{\mathbb Q}/\mathbb Q) \rightarrow \operatorname{Aut}(E[\ell])\) 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 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 \(\ell \in \{11,13,17\}\).
Reviewer: Bernat Plans (Barcelona)
Citations:
Zbl 1028.11035Software:
ecdataReferences:
| [1] | Auer, R.; Top, J., Legendre elliptic curves over finite fields, J. Number Theory, 95, 2, 303-312 (2002) · Zbl 1081.11044 |
| [2] | Birch, B., Noncongruence subgroups, covers and drawings, (Schneps, Leila, The Grothendieck Theory of Dessins d’Enfants (1994), Cambridge Univ. Press), 25-46 · Zbl 0930.11024 |
| [3] | Brillhart, J.; Morton, P., Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial, J. Number Theory, 106, 1, 79-111 (2004) · Zbl 1083.11036 |
| [4] | Carlitz, L., Congruence properties of special elliptic functions, Monatsh. Math., 58, 77-90 (1954) · Zbl 0058.27305 |
| [5] | Cremona, J. E., Algorithms for Modular Elliptic Curves (1992), Cambridge Univ. Press · Zbl 0758.14042 |
| [6] | Elkies, N. D., The existence of infinitely many supersingular primes for every elliptic curve over \(Q\), Invent. Math., 89, 3, 561-567 (1987) · Zbl 0631.14024 |
| [7] | Klüners, J.; Malle, G., Counting nilpotent Galois extensions, J. Reine Angew. Math., 572, 1-26 (2004) · Zbl 1052.11075 |
| [8] | Mazur, B., Rational isogenies of prime degree, Invent. Math., 44, 129-162 (1978) · Zbl 0386.14009 |
| [9] | Plans, B., Central embedding problems, the arithmetic lifting property, and tame extensions of \(Q\), Int. Math. Res. Not., 23, 1249-1267 (2003) · Zbl 1044.12004 |
| [10] | Plans, B.; Vila, N., Tame \(A_n\)-extensions of \(Q\), J. Algebra, 266, 1, 27-33 (2003) · Zbl 1057.12003 |
| [11] | Reverter, A.; Vila, N., Images of mod \(p\) Galois representations associated to elliptic curves, Canad. Math. Bull., 44, 3, 313-322 (2001) · Zbl 1028.11035 |
| [12] | Serre, J.-P., Groupes de Lie \(ℓ\)-Adiques Attachés aux Courbes Elliptiques, (Colloque de Clermont-Ferrand (1964), IHES) · Zbl 0148.41502 |
| [13] | Serre, J.-P., Proprietes galoisiénnés des points d’ordre fini des courbes elliptiques, Invent. Math., 15, 259-331 (1972) · Zbl 0235.14012 |
| [14] | Serre, J.-P., Oeuvres, vol. III, 1972-1984 (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0849.01049 |
| [15] | Serre, J.-P., Abelian \(ℓ\)-Adic Representations and Elliptic Curves (1989), Addison-Wesley Publishing Company · Zbl 0709.14002 |
| [16] | Silverman, J., The Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 106 (1986), Springer · Zbl 0585.14026 |
| [17] | Weber, H., Lehrbuch der Algebra III (1908), Vieweg: Vieweg Braunschweig |
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