Related papers: The Langlands-Shahidi method over function fields:…

We apply the method of multiple Dirichlet series to develop $L$-functions ratios conjecture with one shift in both the numerator and denominator in certain ranges for the family of quartic Hecke $L$-functions of prime moduli over the…

We prove a conjecture of Prasad and Takloo-Bighash for discrete series of inner forms of the general linear group over a non archimedean local field, in the case of Steinberg representations.

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the…

We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where $F$ is a non-archimedean local field of characteristic zero.

Let $F$ be a non-archimedean local field and $G={\bf{G}}(F)$ the group of $F$-rational points of a connected reductive $F$-group. Then we have the Langlands classification of complex irreducible admissible representations $\pi$ of $G$ in…

The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied…

We study conjectures of Ben-Zvi--Sakellaridis--Venkatesh that categorify the relationship between automorphic periods and $L$-functions in the context of the Geometric Langlands equivalence. We provide evidence for these conjectures in some…

In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi Conjecture given previous results on the…

Global internal symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to extended observables by considering unitary actions of finite global 2-group symmetries…

The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be…

The recent proposal by Ben-Zvi, Sakellaridis and Venkatesh of a duality in the relative Langlands program, leads, via the process of quantization of Hamiltonian varieties, to a duality theory of branching problems. This often unexpectedly…

We establish the functorial transfer of generic, automorphic representations from the quasi-split general spin groups to general linear groups over arbitrary number fields, completing an earlier project. Our results are definitive and, in…

In this paper we first describe the method of field patching, developed by Harbater and Hartmann, paying special attention to the relationship between factorization and local-global principles, and second, we extend the basic factorization…

The Rankin-Selberg method for studying Langlands' automorphic $L$-functions is to find integral representations, involving certain Fourier coefficients of cusp forms and Eisenstein series, for these functions. In this thesis we develop the…

We propose a strengthening of the Grothendieck--Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities. Version 2: Main…

Based on results obtained in a companion paper [MSRI preprint 1997-002], we construct groups of special $S$--units for function fields of characteristic $p>0$, and show that they satisfy Gras--type Conjectures. We use these results in order…

First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…

We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.

We study the local zeta integrals attached to a pair of generic representations $(\pi,\tau)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions…

We provide a motivated introduction to the theory of categorical actions of groups and the local geometric Langlands program. Along the way we emphasize applications, old and new, to the usual representation theory of reductive and affine…