Nicolas Templier

Hybrid sup-norm bounds for Hecke–Maass cusp forms

Let f be a Hecke–Maass cusp form of eigenvalue λ and square-free level N . Normalize the hyperbolic measure such that vol(Y0(N)) = 1 and the form f such that ‖f‖ 2 = 1. It is shown that ‖f‖ ∞ ≪ǫ λ 5 24 N 1 3 +ǫ for all ǫ > 0. This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

Families of L -Functions and Their Symmetry

A few years ago the first-named author proposed a working definition of a family of automorphic L-functions. Then the work by the second and third-named authors on the Sato–Tate equidistribution for families made it possible to give a conjectural answer for the universality class introduced by Katz and the first-named author for the distribution of the zeros near s = 1/2. In this article we develop these ideas fully after introducing some structural invariants associated to the arithmetic statistics of a family.

ON FIELDS OF RATIONALITY FOR AUTOMORPHIC REPRESENTATIONS

This paper proves two results on the field of rationality Q(�) for an automorphic representa- tion �, which is the subfield of C fixed under the subgroup of Aut(C) stabilizing the isomorphism class of the finite part of �. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representationssuch thatis unramified away from a fixed finite set of places, �1 has a fixed infinitesimal character, and (Q(�) : Q) is bounded. The second main result is that for classical groups, (Q(�) : Q) grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed L-packet under mild conditions.

Heegner points and Eisenstein series

Abstract We give an alternative computation of the twisted second moment of critical values of class group L-functions attached to an imaginary quadratic field. The method avoids long calculations and yields the polynomial growth in the s-parameter for the remaining term.

Asymptotics and Local Constancy of Characters of p-adic Groups

In this paper we study quantitative aspects of trace characters \(\Theta _{\pi }\) of reductive p-adic groups when the representation π varies. Our approach is based on the local constancy of characters and we survey some other related results. We formulate a conjecture on the behavior of \(\Theta _{\pi }\) relative to the formal degree of π, which we are able to prove in the case where π is a tame supercuspidal. The proof builds on J.-K. Yu’s construction and the structure of Moy–Prasad subgroups.

Sur le rang des courbes elliptiques sur les corps de classes de Hilbert

Let E /ℚ be an elliptic curve and let D contains Heegner points of discriminant D , those points generate a subgroup of rank at least | D | δ , where δ >0 is an absolute constant. This result is compatible with the Birch and Swinnerton-Dyer conjecture.