Algebra & Number Theory

Torsion des modules de Drinfeld de rang 2 et formes modulaires de Drinfeld

Cécile Armana

Full-text: Open access

Abstract

On donne des résultats de non-existence pour les points rationnels de la courbe modulaire de Drinfeld affine Y1(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 q7. 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 Y1(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 q7. 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.

Article information

Source
Algebra Number Theory, Volume 6, Number 6 (2012), 1239-1288.

Dates
Received: 14 July 2011
Revised: 8 November 2011
Accepted: 10 December 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729867

Digital Object Identifier
doi:10.2140/ant.2012.6.1239

Mathematical Reviews number (MathSciNet)
MR2968640

Zentralblatt MATH identifier
1297.11056

Subjects
Primary: 11G09: Drinfelʹd modules; higher-dimensional motives, etc. [See also 14L05]
Secondary: 11F52: Modular forms associated to Drinfelʹd modules 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 14G05: Rational points

Keywords
torsion of Drinfeld modules Drinfeld modular forms Drinfeld modular curves

Citation

Armana, Cécile. Torsion des modules de Drinfeld de rang 2 et formes modulaires de Drinfeld. Algebra Number Theory 6 (2012), no. 6, 1239--1288. doi:10.2140/ant.2012.6.1239. https://projecteuclid.org/euclid.ant/1513729867


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