On donne des résultats de non-existence pour les points rationnels de la courbe modulaire de Drinfeld affine Y 1 (p) avec p idéal premier de ‫ކ‬ q [T ]. Cette courbe classifie les modules de Drinfeld de rang 2 munis d'un point de torsion d'ordre p. Le premier énoncé concerne les points définis sur les extensions de ‫ކ‬ q (T ) quadratiques pour p de degré 3 et cubiques pour p de degré 4 et q ≥ 7. Le deuxième, conditionné à une dualité entre algèbre de Hecke et formes modulaires de Drinfeld, concerne les points sur les extensions de degré ≤ q pour deg p suffisamment grand. Comme conséquence, on déduit, sous la même condition, une borne uniforme pour la torsion des modules de Drinfeld de rang 2 définis sur les extensions de ‫ކ‬ q (T ) de degré ≤ q, prédite par Poonen.We give nonexistence results for rational points on the affine Drinfeld modular curve Y 1 (p) with p a prime ideal of ‫ކ‬ q [T ]. This curve classifies Drinfeld modules of rank 2 with a torsion point of order p. The first statement concerns points defined over quadratic extensions of ‫ކ‬ q (T ) for p of degree 3 and cubic extensions of ‫ކ‬ q (T ) for p of degree 4 and q ≥ 7. The second statement is valid under a duality condition between Hecke algebra and Drinfeld modular forms, and concerns points over extensions of degree ≤ q whenever deg p is sufficiently large. As a consequence we derive, under the same condition, a uniform bound for the torsion of rank-2 Drinfeld modules defined over extensions of ‫ކ‬ q (T ) of degree ≤ q, as predicted by Poonen.

Consider the space of Drinfeld modular forms of fixed weight and type for Γ 0 (n) ⊂ GL 2 (F q [T ]). It has a linear form b n , given by the coefficient of t m+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b 1 . We give elements in the Hecke annihilator of b 1 . Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients b n , where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.

On donne des résultats de non-existence pour les points rationnels de la courbe modulaire de Drinfeld affine Y 1 (p) avec p idéal premier de ‫ކ‬ q [T ]. Cette courbe classifie les modules de Drinfeld de rang 2 munis d'un point de torsion d'ordre p. Le premier énoncé concerne les points définis sur les extensions de ‫ކ‬ q (T ) quadratiques pour p de degré 3 et cubiques pour p de degré 4 et q ≥ 7. Le deuxième, conditionné à une dualité entre algèbre de Hecke et formes modulaires de Drinfeld, concerne les points sur les extensions de degré ≤ q pour deg p suffisamment grand. Comme conséquence, on déduit, sous la même condition, une borne uniforme pour la torsion des modules de Drinfeld de rang 2 définis sur les extensions de ‫ކ‬ q (T ) de degré ≤ q, prédite par Poonen.We give nonexistence results for rational points on the affine Drinfeld modular curve Y 1 (p) with p a prime ideal of ‫ކ‬ q [T ]. This curve classifies Drinfeld modules of rank 2 with a torsion point of order p. The first statement concerns points defined over quadratic extensions of ‫ކ‬ q (T ) for p of degree 3 and cubic extensions of ‫ކ‬ q (T ) for p of degree 4 and q ≥ 7. The second statement is valid under a duality condition between Hecke algebra and Drinfeld modular forms, and concerns points over extensions of degree ≤ q whenever deg p is sufficiently large. As a consequence we derive, under the same condition, a uniform bound for the torsion of rank-2 Drinfeld modules defined over extensions of ‫ކ‬ q (T ) of degree ≤ q, as predicted by Poonen.

International audienceGiven a prime number l greater than or equal to 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 \rho_{A, l}: Gal_Q -> GSp(6, l) 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 Sp(6, l)