Une base explicite de symboles modulaires sur les corps de fonctions

Modular symbols for the congruence subgroup $\Gamma_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin's for classical modular symbols. We completely solve the relations and get an explicit basis of generators when $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for $L$-functions of certain automorphic cusp forms for $\mathbf{F}_q(T)$. The main statement also provides a key-step for a result towards the uniform boundedness conjecture for Drinfeld modules of rank $2$.Comment: Journal f\"ur die reine und angewandte Mathematik (2014) \`a para\^itre, in Frenc

Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].Comment: Minor changes. 13 pages. This paper contains results of the collaboration started at the conference Women in numbers - Europe, (October 2013), by the working group "Galois representations and Galois groups over Q

Sur les symboles modulaires de Manin-Teitelbaum pour Fq(T)

"Vegeu el resum a l'inici del document del fitxer adjunt"

Coefficients of Drinfeld modular forms and Hecke operators

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Large Galois images for Jacobian varieties of genus 3 curves

Given a prime number ℓ ≥ 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation ρA,ℓ : GQ → GSp6 (Fℓ) attached to the ℓ-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp6 (Fℓ).Ministerio de Economía y CompetitividadUniversité de Franche-ComtéAgence nationale de la recherch

Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve, let J(C) be the associated Jacobian variety and let ¯ρℓ : GQ → GSp(J(C)[ℓ]) be the Galois representation attached to the ℓ-torsion of J(C). Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes ℓ (if they exist) such that ¯ρℓ is surjective. In particular we realize GSp6 (Fℓ) as a Galois group over Q for all primes ℓ ∈ [11, 500000]

COEFFICIENTS OF DRINFELD MODULAR FORMS AND HECKE OPERATORS

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Torsion rationnelle des modules de Drinfeld

This thesis studies the existence of torsion points of rank 2 Drinfeld modules over finite extensions of F_q(T) (q a prime power). Following the approach of Mazur and Merel for the torsion of elliptic curves over number fields, we introduce and study a quotient of the Jacobian of a Drinfeld modular curve, defined by a special Teitelbaum modular symbol. Under a hypothesis of duality between Hecke algebra and modular forms for F_q[T], and a minor technical hypothesis, we show the following result: if there exists a rank 2 Drinfeld module over an extension of degree at most q of F_q(T), with a torsion point of order a prime ideal n of F_q[T], then the degree of n is at most max(q,4). For this purpose, we use a description of the action of the Hecke algebra on Teitelbaum modular symbols and on modular forms for F_q[T]. When n has small degree, we obtain unconditional results: there exists no rank 2 Drinfeld module over an extension of degree at most 2 (resp. at most 3) of F_q(T) with a torsion point of order a prime ideal of degree 3 (resp. 4 if q is at least 7). These statements partially confirm a conjecture of Poonen and Schweizer, about a uniform bound for the torsion of Drinfeld modules.Cette thèse étudie l'existence de points de torsion pour les modules de Drinfeld de rang 2 sur des extensions finies de F_q(T), pour q puissance d'un nombre premier. Notre approche suit celle de Mazur et Merel pour la torsion des courbes elliptiques sur les corps de nombres : nous introduisons un quotient de la jacobienne d'une courbe modulaire de Drinfeld, défini à l'aide d'un symbole modulaire de Teitelbaum particulier, et étudions ses propriétés. Sous une hypothèse de dualité entre algèbre de Hecke et formes modulaires pour F_q[T], ainsi qu'une hypothèse technique mineure, on montre le résultat suivant : s'il existe un module de Drinfeld de rang 2 sur une extension de degré au plus q de F_q(T), muni d'un point de torsion d'ordre un idéal premier n de F_q[T], alors le degré de n est au plus max(q,4). Nous utilisons pour cela une description de l'action de l'algèbre de Hecke sur les symboles modulaires de Teitelbaum et sur les formes modulaires pour F_q[T]. Lorsque le degré de n est petit, on obtient des résultats non conditionnels : il n'existe aucun module de Drinfeld de rang 2 sur une extension de degré au plus 2 (resp. au plus 3) de F_q(T) possédant un point de torsion d'ordre un idéal premier de degré 3 (resp. de degré 4 si q est au moins 7). Cela confirme partiellement une conjecture de Poonen et Schweizer de borne uniforme sur la torsion des modules de Drinfeld