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

Abstract

Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rank of certain abelian varieties defined over É. 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 É by Kowalski, Michel and VanderKam. In this setting, there is an additional term, coming from old forms, to control. Finally we convert our estimates for the harmonic moments into ones for the natural moments.

Non-vanishing of Rankin-Selberg convolutions for Hilbert modular forms

“…A brief account on Hilbert modular forms can be found in a recent work of the authors [11, Section 1.2] or Trotabas [22,Section 3]. However, for a more detailed exposition on the topic, the reader is referred to Garrett Given two primitive forms f ∈ S k (O F ) and g ∈ S k (n), one defines the L-series for the Rankin-Selberg convolution of f and g as…”

“…A standard application of an approximate functional equation and a Petersson trace formula (see [22,Proposition 6.3]) allows us to express (4.1) as…”

Determining Hilbert modular forms by central values of Rankin–Selberg convolutions: the weight aspect

“…In this section, we recall the definition and some properties of the space of adèlic Hilbert modular forms, and we explain briefly the relation it bears to the space of classical Hilbert modular forms. Our exposition in the adèlic setting borrows heavily from that of Trotabas [17,Section 3].…”

“…Here x is the unique element in (a −1 D F c/a −1 D F cc 2 ) × such that xx ≡ 1 mod cc. The reader is referred to Section 2.2 and Section 6 in [17] for more details on this construction.…”

Determining Hilbert modular forms by central values of Rankin-Selberg convolutions: The level aspect

“…We now fix some notation pertaining to the space of adèlic Hilbert modular forms. To this end, we closely follow the exposition in [17]. Let A F be the adèle ring of F .…”