Non annulation des fonctions $L$ des formes modulaires de Hilbert au point central

Trotabas, Denis

Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 187-259.

La conjecture de Birch et Swinnerton-Dyer donne des estimations fines sur le rang de certaines variétés abéliennes définies sur $\mathbf{Q}$. Dans le cas des jacobiennes des courbes modulaires, ce problème est équivalent à l’estimation de l’ordre d’annulation en $1/2$ des fonctions $L$ des formes modulaires, et a été traité inconditionnellement par Kowalski, Michel et VanderKam. L’objet de ce travail est d’étendre cette approche dans le cas d’un corps totalement réel arbitraire, ce qui nécessite l’utilisation de la théorie adélique. Nous suivons la méthode des moments amolis initiée par Selberg. On généralise la formule de Petersson que l’on utilise pour étudier les deux premiers moments harmoniques, ce qui nous permet d’atteindre inconditionnellement les mêmes proportions de formes dont la fonction $L$ est non nulle en $1/2$ que celles établies pour $\mathbf{Q}$. Dans cette situation, il y a un terme additionnel, issu des formes anciennes, à contrôler.

Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rank of certain abelian varieties defined over $\mathbf{Q}$. in the case of the jacobian of the modular curves, this problem is equivalent to the estimation of the order of vanishing at $1/2$ of $L$-functions of classical modular forms, and was treated, without assuming the Riemann hypothesis, by Kowalski, Michel and VanderKam. The purpose of this paper is to extend this approach in the case of an arbitrary totally real field, which necessitates an appeal of Jacquet-Langlands’ theory and the adelization of the problem. To show that the $L$-function (resp. its derivative) of a positive density of forms does not vanish at $1/2$, we follow Selberg’s method of mollified moments (Iwaniec, Sarnak, Kowalski, Michel and VanderKam among others applied it successfully in the case of classical modular forms). We generalize the Petersson formula, and use it to estimate the first two harmonic moments, this then allows us to match the same unconditional densities as the ones proved over $\mathbf{Q}$ by Kowalski, Michel and VanderKam. In this setting, there is an additional term, coming from old forms, to control.

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