Related papers: Kuznetsov's trace formula and the Hecke eigenvalue…

We establish an infinitesimal variant of Guo-Jacquet trace formula for the case of a central simple algebra over a number field $F$ containing a quadratic field extension $E/F$. It is an equality between a sum of geometric distributions on…

We show that with respect to the q-Plancherel measure on partitions of size n, the irreducible characters of an Hecke algebra $H_q(S_n)$ are concentrated around the normalized trace of $H_q(S_n)$. More precisely, we prove that the…

Let $A(1,m)$ be the Fourier coefficients of a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $\pi_1$ and $\lambda(m)$ be those of a $SL(2,\mathbb{Z})$ Hecke holomorphic or Hecke-Mass cusp form $\pi_2$. Let $H\subset[\![…

A formula of Kuznetsov allows one to interpret a smooth sum of Kloosterman sums as a sum over the spectrum of $GL(2)$ automorphic forms. In this paper, we construct a similar formula for the first hyper-Kloosterman sums using $GL(3)$…

Let $f$ be a normalized holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k$ with $k\equiv0\bmod 4$. By the Kuznetsov trace formula for $GL_3(\mathbb R)$, we obtain the first moment of central values of $L(s,f\otimes \phi)$, where…

We define a unified trace form $\tau$ on the cyclotomic Hecke algebras $\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $\xi\neq 1$) and Brundan-Kleshchev's…

Traces of singular moduli can be approximated by exponential sums of quadratic irrationals. Recently Andersen and Duke used theory of Maass forms to estimate generalized twisted traces with power-saving error bounds. We establish an…

Let $E/F$ be an extension of number fields with $\mathrm{Gal}(E/F)$ simple and nonabelian. In [G] the first named author suggested an approach to nonsolvable base change and descent of automorphic representations of $\mathrm{GL}_2$ along…

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are…

We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and…

Let O^1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indefinite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the…

We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$…

In this note, we derive a relative trace formula (RTF) using classical methods. We obtain a closed formula for the second moment of the central values of holomorphic cusp forms, a result originally established in Kuznetsov's preprint.

We assemble three basic analytic inputs -- the Kuznetsov trace formula on $\mathrm{SL}_2(\mathbb Z)$ with explicit continuous spectrum, the $\mathrm{GL}_3$ Voronoi formula, and $t$-aspect second-moment bounds for $L(1/2+it,\varphi)$ -- into…

Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application,…

Let $\lambda_{\pi}(m,n)$ be the Fourier coefficients of a Hecke-Maass cusp form $\pi$ for $SL(3,\mathbb{Z})$ and $\lambda_{f}(n)$ be the Fourier coefficients of Hecke-eigen form $f$ for $SL(2,\mathbb{Z})$. The aim of this article is to get…

We introduce a framework within which a large class of joint equidistribution problems can be studied and resolved with effective error terms. This involves proving a higher dimensional and $\mu$-analogue of the Erd\"{o}s-Tur\'{a}n…

We prove a Burgess-like subconvex bound for twisted L-functions of a fixed irreducible cuspidal automorphic representation of GL(2) over a totally real number field. The proof is based on a spectral decomposition of shifted convolution sums…

We prove $\sum_{K-G}\le \kappa_j \le K+G}\alpha_j H_j^3(1/2) \ll_\epsilon GK^{1+\epsilon}$ for $K^\epsilon \le G \le K$, where $\alpha_j = |\rho_j(1)|^2(\cosh \pi\kappa_j)^{-1}$, and $\rho_j(1)$ is the first Fourier coefficient of the Maass…

We study a mean value of the shifted convolution problem over the Hecke eigenvalues of a fixed non-holomorphic cusp form. We attain a result also for a weighted case. Furthermore, we point out that the proof yields analogous upper bounds…