Étienne Fouvry

On the distribution of supersingular primes

Let E be a fixed elliptic curve defined over the rational numbers. We prove that the number of primes p < x such that E has supersingular reduction modp is greater than 0og4*)* for any positive 6 and x sufficiently large. Here log^x is defined recursively as log(log .̂_1 JC) and log! x = log*. We also establish several results related to the LangTrotter conjecture.

Low-lying zeros of dihedral L-functions

Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the L-functions L (s, ψ), whereψ is a character of the ideal class group of the imaginary quadratic fieldQ( √ −D) (D squarefree, D> 3, D ≡ 3 (mod 4)). We prove that, in the vicinity of the central point s = 1/2, the average distribution of these zeros (for D−→ ∞) is governed by the symplectic distribution. By averaging over D, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes p of the form4p = m2 + Dn2, and sieve techniques are applied.

New equidistribution estimates of Zhang type

In May 2013, Y. Zhang [52] proved the existence of infinitely many pairs of primes with bounded gaps. In particular, he showed that there exists at least one h ě 2 such that the set tp prime | p` h is primeu is infinite. (In fact, he showed this for some even h between 2 and 7ˆ 10, although the precise value of h could not be extracted from his method.) Zhang’s work started from the method of Goldston, Pintz and Yıldırım [23], who had earlier proved the bounded gap property, conditionally on distribution estimates concerning primes in arithmetic progressions to large moduli, i.e., beyond the reach of the Bombieri–Vinogradov theorem. Based on work of Fouvry and Iwaniec [11, 12, 13, 14] and Bombieri, Friedlander and Iwaniec [3, 4, 5], distribution estimates going beyond the Bombieri–Vinogradov range for arithmetic functions such as the von Mangoldt function were already known. However, they involved restrictions concerning the residue classes which were incompatible with the method of Goldston, Pintz and Yıldırım. Zhang’s resolution of this difficulty proceeded in two stages. First, he isolated a weaker distribution estimate that sufficed to obtain the bounded gap property (still

Algebraic Trace Functions Over The Primes

We study sums over primes of trace functions of l-adic sheaves. Using an extension of our earlier results on algebraic twists of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann hypothesis over finite fields, we prove general estimates with power saving for such sums. We then derive various concrete applications.

Sur certaines sommes d'exponentielles sur les nombres premiers

Keywords: exponential sums ; sums over prime numbers Reference TAN-ARTICLE-1998-001 Record created on 2008-11-14, modified on 2017-05-12

The Square Sieve and the Lang-Trotter Conjecture

Let E be an elliptic curve dened over Q and without complex multiplication. Let K be a xed imaginary quadratic eld. We nd nontrivial upper bounds for the number of ordinary primes p x for which Q( p) = K; where p denotes the Frobenius endomorphism of E at p: More precisely, under a certain generalized Rie- mann hypothesis we show that this number is OE x 17 18 log x ; and unconditionally

On The Exponent Of Distribution Of The Ternary Divisor Function

We show that the exponent of distribution of the ternary divisor function d(3) in arithmetic progressions to prime moduli is at least 1/2 + 1/46, improving results of Friedlander-Iwaniec and Heath-Brown. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to 1/2 + 1/34.

A study in sums of products

We give a general version of cancellation in exponential sums that arise as sums of products of trace functions satisfying a suitable independence condition related to the Goursat–Kolchin–Ribet criterion, in a form that is easily applicable in analytic number theory.

On moments of twisted L-functions

We study the average of the product of the central values of two

Rang des courbes elliptiques et sommes d'exponentielles

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