Related papers: Primitives of volume forms in Carnot groups

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in R^{n+1} is close to a constant in L^{n}-sense, then the set is close to a union of disjoint balls with respect to the…

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…

Let $X$ be an $n$-dimensional manifold and $V_1, \ldots, V_n \subset C^\infty(X, \mathbb R)$ finite-dimensional vector spaces with Euclidean metric. We assign to each $V_i$ a Finsler ellipsoid, i.e., a family of ellipsoids in the fibers of…

Let $E/\mathbb{Q}$ be a number field of degree $n$. We show that if $\operatorname{Reg}(E)\ll_n |\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central…

Let $\pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $\pi$ has a generic global Arthur parameter, we establish the…

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset…

A quadratic twist of the L-function associated with a modular form is known to satisfy a functional equation, which may be even or odd. A result due to Gross and Zagier explicitly computes the central value of the L-function or its…

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such…

Let $G$ be a simply connected compact Lie group. Let $L_e(G)$ be the based loop group with the base point $e$ which is the identity element. Let $\nu_e$ be the pinned Brownian motion measure on $L_e(G)$ and let $\alpha\in…

Let X_1,...,X_q be the basis of the space of horizontal vector fields on a homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system of N equations, in divergence form, structured on these vector fields, where the…

We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused…

The volume of a Cartier divisor is an asymptotic invariant, which measures the rate of growth of sections of powers of the divisor. It extends to a continuous, homogeneous, and log-concave function on the whole N\'eron--Severi space, thus…

We prove that if the cardinality of a subset of the 2-dimensional vector space over a finite field with $q$ elements is $\ge \rho q^2$, with $\frac{1}{\sqrt{q}}<<\rho \leq 1$, then it contains an isometric copy of $\ge c\rho q^3$ triangles.

Let $f$ be a Maass form for $SL(3, \mathbb{Z})$ which is fixed and $u_j$ be an orthonormal basis of even Maass forms for $SL(2, \mathbb{Z}),$ we prove an asymptotic formula for the average of the product of the Rankin-Selberg $L$-function…

Let $\mathbb{F}_{q}$ denote the finite field of order $q$ (a power of a prime $p$). We study the $p$-adic valuations for zeros of $L$-functions associated with exponential sums of the following family of Laurent polynomials…

Let D be a square-free polynomial in F_q[t], where q is odd, and let G be a genus of definite ternary lattices over F_q[t] of determinant D. In this paper we give self-contained and relatively elementary proofs of Siegel's formulas for the…

We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study…

The first and second moments are established for the family of quadratic Dirichlet $L$--functions over the rational function field at the central point $s=\tfrac{1}{2}$ where the character $\chi$ is defined by the Legendre symbol for…

We establish an operator-theoretic uncertainty principle over arbitrary compact groups, generalizing several previous results. As a consequence, we show that if f is in L^2(G), then the product of the measures of the supports of f and its…

Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $\lambda$ of the field of algebraic numbers which is prime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ as an object…