We study, on average over f, zeros of the L-functions of primitive weight two forms of level q (fixed). We prove, on the one hand, density theorems for the zeros (similar to the results of Bombieri, Jutila, Motohashi, Selberg in the case of characters), which are applied in \cite{KM} to obtain a sharp unconditionnal estimate of the (analytic) rank of the new part of J_0(q); and, on the other hand, non-vanishing theorems at the critical point, showing that a positive proportion of L-functions are non zero there.
The question about modular forms have recently received a lot of attention; concerning the non-vanishing of automorphic L-functions Michel, Kowalski and Vanderkam proved (among others results) that there's positive proportion of non-vanishing of primitives forms at the critical point. This result was proved by these authors in the prime level case; on the othe hand, Iwaniec, Luo and Sarnak showed the same result for the squarefree level case. In order to understand the influence of the arithmetic shape of level forms in their vanishing, this paper studies a generalisation to the primitives for…
The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their decompensation on the prime numbers. Chapter 2 is devoted to the results known concerning the first problem in the case of classical modular forms and to explain our motivation of research. In Chapter 3, we treated the case of primitive Maass forms and obtained an estimate of the subconvexity-type for the least integer n such that the n-th coefficient is negat…
Nous étudions la nature arithmétique de $q$-analogues des valeurs $ζ(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $ζ_q(s)= \sum_{k=1} ^{\infty}q^k \sum_{d\mid k} ^{}d^{s-1}$, $s=1,2,...$, o{ù} $q$ est un nombre complexe, $| q|<1$ (ces fonctions sont intimenent liées au monde automorphe). Le théorème principal de cet article montre que, si $1/q$ est un nombre entier différent de $\pm1$ et si $M$ est un nombre impair suffisamment grand, alors la dimension de l'espace vectoriel engendré sur $\mathbb Q$ par $1,ζ_q(3), ζ_q(5),..., ζ_q(M)$ est au moins $c_1\cdot\sqrt{M}$, …
In this article, we determine the trace of some Hecke operators on the spaces of level one automorphic forms on the special orthogonal groups of the euclidean lattices $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{E}_8\oplus \mathrm{A}_1$, with arbitrary weight. Using Arthur's theory, we deduce properties of the Satake parameters of the automorphic representations for the linear groups discovered by Chenevier and Renard. Our results corroborate a conjecture by Bergström, Faber and van der Geer about the Hasse-Weil zeta function on the moduli spaces of $17$-pointed curves of genus $3$.
We prove classification results for the cuspidal automorphic algebraic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) of small prime conductor and small motivic weight, in the spirit of the works of Chenevier, Lannes and Taïbi in conductor $1$. The main result is an explicit list of all such representations with motivic weight up to $17$ and conductor $2$. For this, we develop the analytical method based on the Riemann-Weil explicit formulas, and use Arthur's work to relate those representations to classical objects. A key ingredient is a special case of Gross' conjecture re…
We construct in a Sonine Space of entire functions a subspace related to the Riemann zeta function and we show that the quotient contains vectors intrinsically attached to the non-trivial zeros and their multiplicities.
A first encounter with $A(λ) := \int_0^{+\infty}\{λt\}\{t\}t^{-2}dt.$ This text contains the detailed calculations leading to the results of our paper "Etude de l'autocorrelation multiplicative de la fonction "partie fractionnaire"", to appear in the Ramanujan Journal.
We describe the points of diff erentiability of the multiplicative autocorrelation function of the "fractional part" function. In connection with this question, we study series involving the fi rst Bernoulli function, the arithmetical function "number of divisors", and the Gauss map from the theory of continued fractions. A key role is played by a function defi ned in 1933 by J. R. Wilton, similar to the Brjuno function of dynamical systems theory. A unifying theme of our exposition is the use of functional equations involving the Gauss map, allowing us to reprove and re fine a theorem of Wilt…
We use Möbius inversion and the Bernoulli polynomials to prove inequalities between the logarithmic summatory function of the Möbius function and weighted averages of its ordinary summatory function.
In this note, following results from Henri Cohen and Winfried Kohnen, we show that for all integer k greater than 12 and divisible by 4, there exists a cuspidal eigenform of weight k for the full modular group SL2(Z) such that its Hecke L-function does not vanish on k/2.
We give a new proof of an approximate functional equation, due to J. R. Wilton, for a trigonometric sum involving the divisor function. This allows us to improve on Wilton's error term and to give an explicit formula for an unspecified function involved in the functional equation.
Let $α\in]0,1]$ and let $Q_j$ $(1\leqslant j\leqslant r)$ denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the $Q_j(n)$ for $x<n\leqslant x+x^α$ is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the $Q_j$.
Let $X$ be a genus $d$ curve with $d\geq 2$ defined over a global function field $K$ of characteristic $p>0$ with $p>2d+1$. Suppose $X$ non-isotrivial. Let $Γ$ be a sub-group of $J(K_s)$, where $J$ is the jacobian of $X$ and $K_s$ is a separable closure of $K$, verifying $Γ/p Γ$ finite. Then one shows that $X\cap Γ$ has finite cardinal and one provides an explicit upper bound. This generalizes a result of Buium and Voloch.
For integer $n\geqslant 1$ and real number $z\geqslant 1$, define $M(n,z):=\sum_{d|n,\,d\leqslant z}μ(d)$ where $μ$ denotes the Möbius function. Put ${\cal L}(y):=\exp\left\{(\log y)^{3/5}/(\log_2y)^{1/5}\right\}$ $(y\geqslant 3)$. We show that, for a suitable, explicit, constant $L>0$ and some absolute $c>0$, we have $S(x,z)= Lx+O\left({x/{\cal L}(3ξ)^c}\right)$ uniformly for $x\geqslant 1$, $ξ\leqslant z\leqslant x/ξ$.
In this article, we give an explicit description of the invertible functions on the Drinfeld symmetric space over $K$ a finite extension of $\mathbb{Q}_p$. We identify them with some distribution spaces over the profinite set of $K$-rationnal points of the projective space. The strategy consists of constructing a map from these distributions to the invertible functions following the methods of Schneider-Stuhler, Iovita-Spiess, de Shalit. We show that it is compatible with the isomorphisms they constructed to compute étale and de Rham cohomology in degree $1$ and that this property forces our d…
Modular symbols for the congruence subgroup $Γ_0(\mathfrak{n})$ of $GL_{2}(\mathbf{F}_q[T])$ have been defined by Teitelbaum. They have a presentation given by a finite number of generators and relations, in a formalism similar to Manin's for classical modular symbols. We completely solve the relations and get an explicit basis of generators when $\mathfrak{n}$ is a prime ideal of odd degree. As an application, we give a non-vanishing statement for $L$-functions of certain automorphic cusp forms for $\mathbf{F}_q(T)$. The main statement also provides a key-step for a result towards the uniform…
Let $(f_1,f_2)$ a $2$-tuple of arithmetic functions and $$Δ_3(n,f_1,f_2):=\sup\limits_{\substack{(u_1,u_2) \in \mathbb{R}^{2} \\(v_1,v_2) \in [0,1]^{2}}}\Big\lvert \sum\limits_{\substack{d_1 d_{2} \mid n \\ e^{u_i}<d_i\leqslant e^{u_i+v_i}}}{f_1(d_1) f_{2}(d_{2})} \Big\rvert{\rm .}$$ In this paper, we give an upper bound of the second moment of $Δ_3(n,χ_1,χ_2)$ when $χ_1$ and $χ_2$ are two non principal Dirichlet characters, following methods developed by La Bretèche and Tenenbaum. This upper bound is a main step for the asymptotic counting of the number of ideals of norm fixed, which will be …
We formulate several variants of a conjecture relating the arithmetic degree of certain hermitian fibre bundles with the values of the logarithmic derivative of Artin's L-functions at negative integers. This generalizes conjectures by Colmez and Gross-Deligne and complements Beilinson's conjectures for the Artin motives. We announce several results in the direction of these statements.
The greatest lower bound of the real parts of the roots of a partial sum of the Dirichlet series of Riemann's zeta function is asymptotically equivalent to the opposite of the number of terms of this sum, multiplied by the Napierian logarithm of 2, when this number of terms tends to infinity.
On the rank of Jacobians over function fields.} Let $f:\mathcal{X}\to C$ be a projective surface fibered over a curve and defined over a number field $k$. We give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the jacobian of the generic fibre (modulo the constant part) in terms of average of the traces of Frobenius on the fibers of $f$. The results also give a reinterpretation of the Tate conjecture for the surface $\mathcal{X}$ and generalizes results of Nagao, Rosen-Silverman and Wazir.
Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de fonctions $K$. On définit l…
In a previous paper, we computed the spectrum of the singular Laplacian attached to canonical metrics on $\mathbb{P}^1$. In this article, we study $ζ_\infty$, the Zeta function associated to this spectrum. We prove that it admits a meromorphic continuation to $\mathbb{C}$. Moreover, this continuation is holomorphic at zero. We compute $ζ_\infty(0)$ and $ζ'_\infty(0)$.
We show a weak form of the function field version of Oesterle's abc conjecture. It asserts that, if $B$ is a complex projective connected curve, the number of intersection points, counted without multiplicities, of a fixed divisor $D$ of degree $d>0$ over $B$ with the graph $H$ of a section $h:B\to B\times \bP^1$ to the first projection is at least $(d-2)n-C(B,D)$, where $n$ is the degree of $H$ over $\bP^1$, and $C(D,B)$ a constant depending only on these two data. We show this number is at least $(d-2[\sqrt {d}]).n-C(D,B)$. The constant is ineffective.
Let $G$ be a connected semisimple group over ${\Bbb Q}$. Given a maximal compact subgroup and a convenient arithmetic subgroup $Γ\subset G({\Bbb Q})$, one constructs an arithmetic manifold $S=S(Γ)=Γ\backslash X$. If $H\subset G$ is a connected, $θ$-stable, semisimple subgroup, for each $g\in G({\Bbb Q})$ one can construct an immersion between arithmetic manifolds $j_g \colon S(H,g)\to S$ induced by the map $H({\Bbb A})\to G({\Bbb A}), h\mapsto gh$. Let us assume that $G$ is anisotropic, which implies that $S$ and $S(H,g)$ are compact. Then, for each positive integer $k$, the map $j_g$ induces …
Nous avons obtenu des formules explicites representant les fonctions E(z) apparaissant dans la theorie des ``Espaces de Sonine'' associes par de Branges a la transformation de Fourier.
Nous étudions la non linéarité des fonctions définies sur F_{2^m} où $m$ est un entier impair, associées aux polynômes de degré 7 ou à des polynômes plus généraux. ----- We study the nonlinearity of the functions defined on F_{2^m} where $m$ is an odd integer, associated to the polynomials of degree 7 or more general polynomials.
Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes, for tori, for groups of multiplicative type, and even for 2-term complexes of tori. We apply these results to the weak approximation for tori over $K$ and to the study of the obstruction to the local-global principle for $K$-torsors under a …
Let $k$ be a field of characteristic 0 and let $K$ be the function field of a smooth projective geometrically integral $k$-curve $X$. Let $T$ be a $K$-torus. In this article, we aim at studying the space of adelic points $T(S,\mathbb{A}_K)$ of $T$ outside a finite set $S$ of closed points of $X$. We start by proving that the group $T(K)$ of rational points of $T$ is always discrete (hence closed) in $T(S,\mathbb{A}_K)$. We then describe the quotient $T(\emptyset,\mathbb{A}_K)/T(K)$ in each of the following three cases: $k$ is an algebraically closed field, $k$ is the field of Laurent series $\…
Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, where they occur in private key systems. In these two cases, the nonlinearity of these function is a main concept. In this article, I show that the spectral amplitude of boolean functions, which is linked to their nonlinearity, is of the order of $2^{m/2}\sqrt{m}$ in mean, whereas its range is bounded by $2^{m/2}$ and $2^m$. Moreover I examine a conjecture of Patterson and Wiedemann saying that the minimum of this spectral amplitude is as close as desired to $2^{m/…
Pour $K$ un corps global (corps de nombres ou corps de fonctions d'une variable sur un corps fini $F$), on dispose de théorèmes de dualité classiques (Tate, Poitou, Nakayama) pour la cohomologie galoisienne à valeurs dans des tores et des modules abéliens finis. Nous établissons de tels théorèmes pour $K$ le corps des fonctions d'une courbe projective et lisse sur le corps $F={\bf C}((t))$ des séries formelles en une variable sur le corps des complexes. Cela permet de contrôler le défaut du principe de Hasse et l'approximation faible pour les espaces homogènes sous un tore. Il y a ici des diff…
Cet exposé est consacré à la preuve de la correspondance de Langlands pour les groupes $\GL_r$ sur les corps de fonctions. ----- This article is devoted to the proof of the Langlands correspondence for the groups $GL_r$ over function fields.
The number of solutions in finite fields of a system of polynomial equations obeys a very strong regularity, reflected for example by the rationality of the zeta function of an algebraic variety defined over a finite field, or the modularity of Hasse-Weil's $L$-function of an elliptic curve over $\Q$. Since two decades, efficient methods have been invented to compute effectively this number of solutions, notably in view of cryptographic applications. This exposé presents some of these methods, generally relying on the use of Lefshetz's trace formula in an adequate cohomology theory and discuss…
Dans un récent article, Lagarias et Soundararajan étudient les solutions friables à l'équation a+b=c. Sous l'hypothèse de Riemann généralisées aux fonctions L de Dirichlet, ils obtiennent une estimation pour le nombre de solutions pondérées par un poids lisse et une minoration pour le nombre de solutions non pondérées. Le but de cet article est de présenter des arguments qui permettent d'une part de préciser les termes d'erreur et d'étendre les domaines de validité de ces estimations afin de faire le lien avec un travail de la Bretèche et Granville, d'autre part d'obtenir le comportement asymp…
Let K be a global field, T a K-torus and S a finite set of places of K. Let K_v be the completion at a place v. Denote by T(O_v) the maximal compact subgroup of the group T(K_v) of K_v-points of T. We show that the diagonal map from T(K) to the product for v in S of the T(K_v)/T(O_v) need not be onto. As a corollary, for suitable v, the group T(O_v) does not cover all R-equivalence classes in T(K_v). While studying the height zeta functions of an algebraic torus over a function field in one variable over a finite field, D. Bourqui showed that Peyre's constant from the number field case should …
