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Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rankof certain abelian varieties defined over $ \Q$. in the case of the jacobian ofthe modular curves, this problem is equivalent to the estimation of the orderof vanishing at 1/2 of $L$-functions of classical modular forms, and wastreated, without assuming the Riemann hypothesis, by Kowalski, Michel andVanderKam. The purpose of this paper is to extend this approach in the case ofan arbitrary totally real field, which necessitates an appeal ofJacquet-Langlands' theory and the adelization of the problem. To show that the$L$-function (resp. its derivative) of a positive density of forms does notvanish at 1/2, we follow Selberg's method of mollified moments (Iwaniec,Sarnak, Kowalski, Michel and VanderKam among others applied it successfully inthe case of classical modular forms). We generalize the Petersson formula, anduse it to estimate the first two harmonic moments, this then allows us to matchthe same unconditional densities as the ones proved over $\Q$ by Kowalski,Michel and VanderKam. In this setting, there is an additional term, coming fromold forms, to control. Finally we convert our estimates for the harmonicmoments into ones for the natural moments.

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