Abstract. We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GLpN q Maass cusp forms for all N 2, satisfy a central limit theorem in a suitable range, generalizing the case N " 2 treated byÉ. Fouvry, S. Ganguly, E. Kowalski and P. Michel in [4]. Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.

We study one-level and two-level densities for low-lying zeros of symmetric power L-functions in the level aspect. This allows us to completely determine the symmetry types of some families of symmetric power L-functions with prescribed sign of functional equation. We also compute the moments of one-level density and exhibit mockGaussian behavior discovered by Hughes & Rudnick.

In [KS16], the authors proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S a, b 0 ; p /p 1/2 converge in the sense of finite distributions to a specific random Fourier series, as a varies over Z/pZ × , b 0 is fixed in Z/pZ × and p tends to infinity among the odd prime numbers. This article considers the case of S a, b 0 ; p n /p n/2 , as a varies over Z/p n Z × , b 0 is fixed in Z/p n Z × , p tends to infinity among the odd prime numbers and n 2 is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on [0, 1] is also established, as (a, b) varies over Z/p n Z × × Z/p n Z × , p tends to infinity among the odd prime numbers and n 2 is a fixed integer. This is the analogue of the result obtained in [KS16] in the prime moduli case.In memory of Kevin Henriot.

No abstract

Abstract. Geometric intuition suggests that the Néron-Tate height of Heegner points on a rational elliptic curve E should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross-Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin-Selberg convolution of E with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted L-function of E by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these L-series and L-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.Reçu par la rédaction le 2 novembre, 2005; revu le 13 juin, 2006. Classification (AMS) par sujet: Primary: 11G50; secondary: 11M41.