Mollification of the fourth moment of automorphic L-functions and arithmetic applications

Abstract

Abstract. We compute the asymptotics of twisted fourth power moments of modular L-functions of large prime level near the critical line. This allows us to prove some new non-vanishing results on the central values of automorphic L-functions, in particular those obtained by base change from GL2(Q) to GL2(K) for K a cyclic field of low degree.

The analytic rank of J0(q) and zeros of automorphic L-functions

“…Moreover, work in progress, in collaboration with J. Vanderkam [KMV1], [KMV2], based on different techniques and much more sophisticated arguments, promises to show that C = 1.18191 is attainable; observe that this is better than Brumer's original bound using the Riemann Hypothesis.…”

Non-vanishing of class group $L$-functions at the central point

“…Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. [1,3,7,9,10,11,13,15,17] and the references given there. This is of interest in various aspects such as the Birch-Swinnerton-Dyer conjecture, the Siegel zero (see [7]) and the theory of modular forms of half-integral weight (see [16,18] [12].…”

Equidistribution, <em>L</em>-functions and ergodic theory: on some problems of Yu. Linnik

“…Finally, from (3.8) one can deduce (3.2), (3.3), (3.4) when π has trivial central character although the derivation may be quite delicate if ϕ2 is an Eisenstein series (cf. [DFI94] and see also [KMV00]). …”