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Theses

Torsion rationnelle des modules de Drinfeld

Abstract : 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.
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https://tel.archives-ouvertes.fr/tel-00338117
Contributor : Cécile Armana <>
Submitted on : Sunday, December 28, 2008 - 6:18:01 PM
Last modification on : Friday, April 10, 2020 - 5:23:36 PM
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  • HAL Id : tel-00338117, version 2

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Cécile Armana. Torsion rationnelle des modules de Drinfeld. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2008. Français. ⟨tel-00338117v2⟩

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