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We reprove the Local Langlands Correspondence for $\GL_n$ over $p$-adic fields as well as the existence of $\ell$-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for…

Let $V$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$. We prove that any continuous linear functional on $V$, which is invariant under the action of the real mirabolic subgroup, is…

Under some technical assumptions of a global nature, we establish the weight part of Serre's conjecture for mod $p$ Galois representations for CM fields that are tamely ramified and sufficiently generic at $p$.

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue…

We show that if two continuous semi-simple \(\ell \)-adic Galois representations are locally potentially equivalent at a sufficiently large set of places then they are globaly potentially equivalent. We also prove an analogous result for…

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$ with Galois automorphism $\sigma$, and let $R$ be an algebraically closed field of characteristic $\ell\notin\{0,p\}$. We…

We describe recent work of Klyachko, Totaro, Knutson, and Tao, that characterizes eigenvalues of sums of Hermitian matrices, and decomposition of tensor products of representations of $GL_n(\mathbb{C})$. We explain related applications to…

We prove in this paper an uniform surjectivity result for Galois representations associated with non-CM $\mathbb{Q}$-curves over imaginary quadratic fields, using various tools for the proof, such as Mazur's method, isogeny theorems,…

We consider the classification problem for compact Lie groups $G\subset U(n)$ which are generated by a single conjugacy class with a fixed number $N$ of distinct eigenvalues. We give an explicit classification when N=3, and apply this to…

We construct small parabolic eigenvarieties for holomorphic Siegel cuspforms of genus $2$ and study families of Galois representations attached to them in the spirit of Bella\"iche--Chenevier. In the course, we introduce the notion of…

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…

We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the…

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…

Let $F$ be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations $Gal(\bar{F}/F) \to PGL_n(C)$ lift to $GL_n(C)$. We take…

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for almost every prime.…

In this paper, we consider mod $\ell$ Galois representations of $\mathbb{Q}$. In particular, we obtain an effective criterion to distinguish two semisimple 2-dimensional, odd mod $\ell$ Galois representations up to isomorphism. Serre's…

We establish the existence of de Rham lifts of Langlands parameters (or Galois representations) for unitary, orthogonal and symplectic (similitude) groups of arbitrary rank. Our results are unconditional except for the assumption $p>2$.

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $\rho$ be an $n$-dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ of regular Hodge-Tate weights. We associate to $\rho$ an explicit…

Let $G$ be a $p$-divisible group over a complete discrete valuation ring $R$ of characteristic $p$. The generic fiber of $G$ determines a Galois representation $\rho$. The image of $\rho$ admits a ramification filtration and a Lie…

We present a collection of conjectural trace identities and explain why they are equivalent to base change and descent of automorphic representations of $\mathrm{GL}_n(\mathbb{A}_F)$ along nonsolvable extensions (under some simplifying…