Related papers: The Local Langlands Conjecture for GSp(4)

Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of…

We prove that every finite dimensional unitary representation of $\mathrm{SL}_{4}(\mathbf{Z})$ contains a non-zero $\mathrm{SL}_{2}(\mathbf{Z})$-invariant vector. As a consequence, there is no sequence of finite-dimensional representations…

Let F be a nonarchimedean locally compact field with residue characteristic p and G(F) the group of F-rational points of a connected reductive group. Following Schneider and Stuhler, one can realize, in a functorial way, any smooth complex…

Let $F$ be a non-Archimedean local field. An irreducible cuspidal representation of $\text{\rm GL}_n(F)$ is epipelagic if its Swan conductor equals 1. We give a full and explicit description of the Langlands parameters of such…

We give a direct proof of the local converse theorem for quasi-split non-split $\mathrm{SO}_{2l}$ over a local non-Archimedean field of characteristic $p\neq 2$, applying the theory of Howe vectors and partial Bessel functions.

Inspired by Emerton's work for GL(2), we study the completed cohomology of the tower of finite sets associated with a definite unitary group in two variables. When p splits (and other technical assumptions are fulfilled), we show that the…

We establish the \emph{inverse conjecture for the Gowers norm over finite fields}, which asserts (roughly speaking) that if a bounded function $f: V \to \C$ on a finite-dimensional vector space $V$ over a finite field $\F$ has large Gowers…

Let $K$ be an algebraically closed field of characteristic zero, $\delta$ a nonzero $\mathcal{E}$-derivation of $K[x]$. We first prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x]$ in some cases. Then we prove that LFED…

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible…

Extending results of Kazhdan to the relative case, we relate harmonic analysis over some spherical spaces G(F)/H(F), where F is a field of positive characteristic, to harmonic analysis over the spherical spaces G(E)/H(E), where E is a…

Let $F$ be a non-archimedean local field of odd residual characteristic. We compute the Jordan set of a simple cuspidal representation of a symplectic group over $F$, using explicit computations of generators of the Hecke algebras of covers…

Let G be a connected reductive group defined over a non-archimedean local field of characteristic 0. We assume G is quasi-split, adjoint and absolutly simple. Let g be the Lie algebra of G. We consider the space of the invariant…

We study the variation of the local Langlands correspondence for ${\rm GL}_{n}$ in characteristic-zero families. We establish an existence and uniqueness theorem for a correspondence in families, as well as a recognition theorem for when a…

We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension E of F. We show that E|F can be chosen to be Galois, after a finite purely inseparable extension of…

We show that local diff-invariant free field theories in four spacetime dimensions do not have local degrees of freedom.

We show that the universal unitary completion of certain locally algebraic representation of $G:=\GL_2(\Qp)$ with $p>2$ is non-zero, topologically irreducible, admissible and corresponds to a 2-dimensional crystalline representation with…

Let $F$ be a non-archimedean local field of odd residual characteristic $p$. The depth of a smooth representation of ${\rm GL}_n(F)$ is an invariant of Local Langlands Correspondence (LLC). The analogous notion on the Galois side of LLC is…

We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials over number fields, assuming a standard conjecture in arithmetic geometry. Our technique simultaneously yields a dynamical…

Let $X$ be a smooth projective curve over a finite field $F_q$. Let $\rho$ be a continuous representation $\pi(X)\to GL_n(F)$, where $F=F_l((t))$ with $F_l$ being another finite field of order prime to $q$. Assume that…

Let $G=Sp(4,\mathbb{R})$ and let $\pi$ be an irreducible, unitary representation of $G$ which is cohomological with respect to trivial coefficients. Using the inclusion from $SO(5,\mathbb{C})$ to $GL(5,\mathbb{C})$, we transfer $\pi$ to an…