Related papers: Global geometrised Rankin-Selberg method for GL(n)

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture…

We show that the local exterior square L-functions of GL_n constructed via the theory of integral representations by Jacquet and Shalika coincide with those constructed by the Langlands-Shahidi method for square integrable representations…

We prove the compatibility at places dividing l of the local and global Langlands correspondences for the l-adic Galois representations associated to regular algebraic essentially (conjugate) self-dual cuspidal automorphic representations…

Let G be a reductive p-adic group. We study how a local Langlands correspondence for irreducible tempered G-representations can be extended to a local Langlands correspondence for all irreducible smooth representations of G. We prove that,…

A theory of graded manifolds can be viewed as a generalization of differential geometry of smooth manifolds. It allows one to work with functions which locally depend not only on ordinary real variables, but also on $\mathbb{Z}$-graded…

We introduce a construction for a Cartan geometry that captures the local behavior of a given geometric automorphism near a distinguished element. The result of this construction, which we call the sprawl generated by the automorphism, is…

In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li…

Studying the analytic properties of the partial Langlands $L$-function via Rankin-Selberg method has been proved to be successful in various cases. Yet in few cases is the local theory studied at the archimedean places, which causes a…

Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global…

We study basic geometric properties of some group analogue of affine Springer fibers and compare with the classical Lie algebra affine Springer fibers. The main purpose is to formulate a conjecture that relates the number of irreducible…

Let $\pi_1,\pi_2$ be irreducible admissible generic tempered representations of $\mathrm{GL}_2(F)$ for some finite extension $F/\mathbf{Q}_p$ of odd residue characteristic. Inspired by work of Loeffler and previous work of the author on…

We study the generalized doubling method for pairs of representations of $G\times GL_k$ where $G$ is a symplectic group, split special orthogonal group or split general spin group. We analyze the poles of the local integrals, and prove that…

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard…

For $E/F$ quadratic extension of local fields and $G$ a reductive algebraic group over $F$, the paper formulates a conjecture classifying irreducible admissible representations of $G(E)$ which carry a $G(F)$ invariant linear form, and the…

I am applying the Langlands-Shahidi method to the metaplectic double cover of Sp(2n). I proved that a Whittaker model of an irreducible admissible representation of this group is unique. As a result I was able to define the local…

Let $X$ be a smooth connected projective algebraic curve over an algebraically closed field, and let $S$ be a finite nonempty closed subset in $X$. We study deformations of $\overline{\mathbb F}_\ell$-sheaves. The universal deformation…

We give a two-variable Rankin--Selberg integral for generic cusp forms on $\mathrm{PGL}_4$ and $\mathrm{PGU}_{2,2}$ which represents a product of exterior square $L$-functions. As a residue of our integral, we obtain an integral…

In this paper, we build the global determinant method of Salberger by Arakelov geometry explicitly. As an application, we study the dependence on the degree of the number of rational points of bounded height in plane curves. We will also…

We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero…

Let $F$ be a local field of characteristic $p>0$. By adapting methods of Scholze, we give a new proof of the local Langlands correspondence for $\GL_n$ over $F$. More specifically, we construct $\ell$-adic Galois representations associated…