Vol. 6, No. 6, 2012

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Torsion des modules de Drinfeld de rang 2 et formes modulaires de Drinfeld

Cécile Armana

Vol. 6 (2012), No. 6, 1239–1288
Abstract

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 Fq[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 Fq(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 degp 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 Fq(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 Fq[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 Fq(T) for p of degree 3 and cubic extensions of Fq(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 degp 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 Fq(T) of degree  q, as predicted by Poonen.

Keywords
torsion of Drinfeld modules, Drinfeld modular forms, Drinfeld modular curves
Mathematical Subject Classification 2010
Primary: 11G09
Secondary: 11F52, 11G18, 14G05
Milestones
Received: 14 July 2011
Revised: 8 November 2011
Accepted: 10 December 2011
Published: 12 August 2012
Authors
Cécile Armana
Max-Planck Institut für Mathematik
Vivatsgasse 7
D-53111 Bonn
Germany