Galois representations and tame Galois realizations

Abstract

[eng] The background of this dissertation is the inverse Galois problem. Which finite groups can occur as Galois groups of an extension of the rational field? This problem was first considered by D. Hilbert, and it still remains open.

Assume that a finite group G can be realized as a Galois group over Q. We can ask whether there exists some other finite Galois extension, with Galois group G and enjoying an additional ramification property. In this connection, several variants of the Inverse Galois Problem have been studied. In this dissertation, we shall address the following problem, posed by Brian Birch around 1994.

Tame Inverse Galois Problem. Given a finite group G, is there a tamely ramified Galois extension K/Q with Galois group G?

In this thesis we address this problem by studying the Galois representations attached to arithmetic-geometric objects such as elliptic curves, or more generally abelian varieties, and modular forms. We seek conditions that ensure that the action of the wild inertia group at all primes is trivial. Note that this strategy of constructing Galois representations such that the image of the wild inertia group at all primes is trivial can be encompassed in the general trend of constructing Galois representations with prefixed local behaviour.

This dissertation is split into two parts. In the first part, we tackle the realization of families of two dimensional linear groups over a finite field as the Galois group of a tamely ramified extension of Q. We study the Galois representations attached to elliptic curves and to modular forms. In the second part we address the problem of realizing a family of four dimensional linear groups over a prime field as the Galois group of a tamely ramified extension of Q. In this part we study the action of the inertia group upon the l-torsion points of the formal group attached to an abelian variety, and obtain a general result that allows us to control the action of the wild inertia group. We apply this result to the formal group attached to abelian surfaces. More precisely, we consider the Jacobians of bielliptic supersingular genus 2 curves, suitably chosen so that we can control the size of the image of the corresponding representation.

The main results we have obtained are the following.

Theorem. Let l be a prime number. There exist infinitely many semistable elliptic curves E with good supersingular reduction at l. The Galois representation attached to the l-torsion points of E provides a tame Galois realization of GL(2, F_l).

Furthermore, we give an explicit algorithm to construct these elliptic curves. The primes l=2, 3, 5, 7 have been considered separately.

Theorem. Let l be a prime number greater than 3. There exist infinitely many genus 2 curves C such that the Galois representation attached to the l-torsion points of the Jacobian of C provides a tame Galois realization of GSp(4, F_l).

As in the previous result, we give an explicit algorithm that enables us to construct these curves.

In addition, we have obtained tame Galois realizations of groups of the form PSL(2, F_(l^2)) for several values of l.

[spa] Esta tesis se desarrolla en torno al Problema Inverso de la Teoría de Galois sobre el cuerpo de los números racionales. Este problema, que fue considerado por primera vez por D. Hilbert, es un problema abierto. En 1994, B. Birch plantea la siguiente variante de este problema, conocida como problema inverso moderado de la teoría de Galois. Dado un grupo finito G, ¿existe una extensión de Galois K/Q, moderadamente ramificada, con grupo de Galois G?

En esta tesis abordamos este problema mediante el estudio de las representaciones de Galois asociadas a objetos aritmético-geométricos, concretamente a curvas elípticas, formas modulares y variedades abelianas. Encontramos condiciones explícitas que garantizan que para todo primo, la imagen del grupo de inercia salvaje es trivial.

La memoria está dividida en dos partes. El objetivo de la primera parte es la obtención de realizaciones moderadas de grupos lineales 2-dimensionales sobre un cuerpo finito como grupos de Galois sobre Q. Dado un número primo l, demostramos que existen infinitas curvas elípticas semiestables E/Q con buena reducción supersingular en l. La representación de Galois asociada a los puntos de l-torsión de E da lugar a una realización de GL(2, F_l) como grupo de Galois de una extensión de Q moderadamente ramificada.

A continuación se consideran las representaciones de Galois asociadas a formas modulares. Obtenemos realizaciones de Galois moderadas para algunos grupos de la familia PSL(2, F_(l^2)).

El objetivo de la segunda parte es la obtención de realizaciones moderadas de los grupos lineales de la familia GSp(4, F_l). Estudiamos la acción de la inercia sobre los puntos de l-torsión del grupo formal asociado a una variedad abeliana, y obtenemos un resultado general que nos permite controlar la acción de la inercia salvaje. Aplicamos este resultado al caso de superficies abelianas. Concretamente, consideramos las Jacobianas de curvas de género 2 bielípticas supersingulares, construidas de forma conveniente para controlar la imagen de la representación asociada. Demostramos que, dado un número primo l mayor que 3, existen infinitas curvas C de género 2 tales que la representación de Galois asociada a los puntos de l-torsión de la Jacobiana de C proporciona una realización de GSp(4, F_l) como grupo de Galois de una extensión moderadamente ramificada de Q.