Related papers: The Siegel variance formula for quadratic forms

We present some aspects of high precision calculations in the context of Lattice Quantum Field Theory. This work is a collection of three studies done during my Ph.D. period. First we present how to use the reweighting technique to…

Let $\lambda_{\phi}(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper…

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\lambda_f(n)$, we…

Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts…

On a probability space $(\Omega, \mathcal F, \mathbb P)$ we consider two independent sequences $(a_k)_{k \geq 1}$ and $(b_k)_{k \geq 1}$ of i.i.d. random variables that are centered with unit variance and which admit a moment strictly…

We give another proof, using tools from Geometric Invariant Theory, of a result due to S. Sam and A. Snowden in 2014, concerning the stability of Kro-necker coefficients. This result states that some sequences of Kronecker coefficients…

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static…

We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums…

We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At…

In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex…

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold with 4-singular fibers. We define the Eichler integrals of the modular forms with half-integral weight, and we show that the invariant is rewritten as a sum…

We determine the asymptotic quantum variance of microlocal lifts of Hecke--Maass cusp forms on the arithmetic compact hyperbolic surfaces attached to maximal orders in quaternion algebras. Our result extends those of Luo--Sarnak--Zhao…

Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm gave an upper bound for modular forms of a given…

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a Hecke congruence subgroup \Gamma =\Gamma_0(q) of the group SL(2,Z[i]), and correspond to exceptional…

Let $f$ and $f'$ be genus $2$ cuspidal Siegel paramodular newforms. We prove that if their Hecke eigenvalues $a_p$ and $a_p'$ satisfy a non-trivial polynomial relation $P(a_p, a_p') = 0$ for a set of primes $p$ of positive density, then $f$…

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up…

Let $k \geq 2$ and $N$ be positive integers and let $\chi$ be a Dirichlet character modulo $N$. Let $f(z)$ be a modular form in $M_k(\Gamma_0(N),\chi)$. Then we have a unique decomposition $f(z)=E_f(z)+S_f(z)$, where $E_f(z) \in…

We prove a local-global principle for representations of binary by quaternary quadratic forms. One of the main ingredients is a recent measure rigidity result of Einsiedler and Lindenstrauss for diagonalizable actions on quotients of…

We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established in [E. Mossel, R. O'Donnell and K. Oleszkiewicz (2010). Noise stability of functions with low…

Let A= (a_{ij}) be a symmetric non-negative integer 2k x 2k matrix. A is homogeneous if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes. Let A be a homogeneous matrix and let F be a general form in C[x_1, \dots x_n] with…