Related papers: Comptage de repr\'esentations cuspidales congruent…

Let F be a non-archimedean local field of characteristic zero with residual characteristic p. In this paper, we present a simple proof and construction of the local Langlands correspondence for simple supercuspidal representations of…

Let $F$ be locally compact field with residue characteristic $p$, and $\mathbf{G}$ a connected reductive $F$-group. Let $\mathcal{U}$ be a pro-$p$ Iwahori subgroup of $G = \mathbf{G}(F)$. Fix a commutative ring $R$. If $\pi$ is a smooth…

Let F be a local field with finite residue field of characteristic p, D the quaternion division algebra with centre F, and R an algebraically closed field of any characteristic. We classify the smooth irreducible R-representations V of the…

If $f$ is a mod-$3$ eigenform of weight 2 and level $\Gamma_0(\ell^2)$ for a prime $\ell$ such that $\ell \equiv -1 \pmod{3}$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but…

We prove classification results for the cuspidal automorphic algebraic representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) of small prime conductor and small motivic weight, in the spirit of the works of Chenevier, Lannes and…

This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet's Lemma for $\mathrm{GL}_2$ in the residually indistinguishable case. We suppose we are given a Galois representation taking values…

Let $F$ be a non-archimedean locally compact field of residue characteristic $p\neq2$, let $G=\mathrm{GL}_{n}(F)$ and let $H$ be an orthogonal subgroup of $G$. For $\pi$ a complex smooth supercuspidal representation of $G$, we give a full…

Let $\rho_\ell$ be a semisimple $\ell$-adic representation of a number field $K$ that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $\rho_\ell$ and completely characterize them, for…

Let $F$ be a locally compact non-archimedean field of residue characteristic $p$, $\textbf{G}$ a connected reductive group over $F$, and $R$ a field of characteristic $p$. When $R$ is algebraically closed, the irreducible admissible…

We show that if $\{\rho_{\ell}\}$ is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation $\overline{\rho}_{\ell}$ is absolutely irreducible for $\ell$ in a density 1 set of…

Let $F$ be the function field of a projective smooth geometrically connected curve $X$ defined over a finite field $\mathbb{F}_q$. Let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Let $S$ be a non-empty finite set of…

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 G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G. At the heart of these conjectures are statements about the geometric structure of Bernstein…

Let G be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex G-representations in the principal series. The parametrization map is injective, and…

Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-adic groups $\{GL_n(F)\}_{n=0}^\infty$, with multiplication defined through parabolic induction. We study the problem of the…

Let G be a reductive p-adic group and let Rep(G)^s be a Bernstein block in the category of smooth complex G-representations. We investigate the structure of Rep(G)^s, by analysing the algebra of G-endomorphisms of a progenerator \Pi of that…

For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner…

Let $\mathbf{G}$ be an unramified quasi-split unitary group over a p-adic field of odd residual characteristic. The goal of this paper is to describe the supercuspidal representations within certain L-packets of $\mathbf{G}$, which are…

Let $G$ be a finite group of Lie type and $\ell$ be a prime which is not equal to the defining characteristic of $G$. In this note we discuss some open problems concerning the $\ell$-modular irreducible representations of $G$. We also…

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier…