Related papers: Integral representation for L-functions for GSp(4)…

We construct an integral transformation intertwining the Gelfand-Tsetlin and the (modified) Gauss-Givental realizations of principle series representations of gl(3). This provides a direct identification of the corresponding integral…

We show the existence of integral models for cuspidal representations of GL(2,q), whose reduction modulo p can be identified with the cokernel of a differential operator on F_{q}[X,Y] defined by J-P. Serre. These integral models come from…

Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We derive estimates for the finite parts of the $L$-functions of irreducible cuspidal $\operatorname{GL}_n({\bf{A}}_F)$-automorphic representations twisted by class…

Motivated by the Langlands' beyond endoscopy proposal for establishing functoriality, we study the representation $\otimes^3$ in a setting related to the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic…

The analytic properties of automorphic L-functions have historically been obtained either through integral representations (the "Rankin-Selberg method"), or properties of the Fourier expansions of Eisenstein series (the "Langlands-Shahidi…

Given a cuspidal automorphic form $\pi$ on $\GL_2$, we study smoothed sums of the form $\sum_{n\in\mathbb{N}} a_{\pi}(n^2+d)W(\frac{n}{Y})$. The error term we get is sharp in that it is uniform in both $d$ and $Y$ and depends directly on…

We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(\pi)$ denote an…

Let $\pi$ be a cohomological automorphic representation of $PGL(2)$ over a number field of arbitrary signature and assume that the local component of $\pi$ at a prime $\mathfrak{p}$ is the Steinberg representation. In this situation one can…

Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual…

In this paper, a construction to imitate the Bernstein-Zelevinsky derivative for automorphic representations on $GL_n(\mathbb{A})$ is introduced. We will later consider the induced representation \[I(\tau_1,\tau_2;\underline{s}) =…

Let $F/F_{0}$ be a quadratic extension of non-archimedean locally compact fields of residue characteristic $p\neq 2$. Let $R$ be an algebraically closed field of characteristic different from $p$. For $\pi$ a supercuspidal representation of…

We prove that the character of an irreducible cuspidal representation of $GL_n(\mathbb{F}_{\ell}((t)))$ is locally bounded up to a logarithmic factor by the orbital integral of a matrix coefficient of this representation. The characteristic…

Let $\pi$ be a regular algebraic cuspidal automorphic representation of ${\rm GL}_n({\mathbb A}_F)$ for a number field $F$. We consider certain periods attached to $\pi$. These periods were originally defined by Harder when $n=2$, and later…

We present a proof of Selberg's Central Limit Theorem for automorphic $L$-functions of degree 2 using Radziwi\l\l\space and Soundararajan's method. Additionally, we prove the independence of the automorphic $L$-functions associated with the…

We prove that Novodvorsky's definition of local L-factors for generic representations of GSp(4) x GL(2) is compatible with the local Langlands correspondence when the GL(2) representation is non-supercuspidal. We also give an interpretation…

In this paper we study the images of certain families $\{\rho_{\pi,\ell} \}_\ell$ of $G_2$-valued Galois representations of $\mbox{Gal}(\overline{F}/F)$ associated to $L$-algebraic regular, self-dual, cuspidal automorphic representations…

Irreducible representations are the building blocks of general, semisimple Galois representations \rho, and cuspidal representations are the building blocks of automorphic forms \pi of the general linear group. It is expected that when an…

Let $\pi$ be an irreducible admissible (complex) representation of $GL(2)$ over a non-archimedean characteristic zero local field with odd residual characteristic. In this paper we prove the equality between the local symmetric square…

Let $G={\rm GL}_{2n}$ over a totally real number field $F$ and $n\geq 2$. Let $\Pi$ be a cuspidal automorphic representation of $G(\mathbb A)$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be…

We are given a finite group $H$, an automorphism $\tau$ of $H$ of order $r$, a Galois extension $L/K$ of fields of characteristic zero with cyclic Galois group $\langle\sigma\rangle$ of order $r$, and an absolutely irreducible…