Related papers: On counting cuspidal automorphic representations f…

We prove an unconditional power saving for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on GL_2 over any number field which is not totally real. Our proof involves the theory of p-adically…

We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description…

We compute the Bessel models of irreducible representations of the finite group $GSp(4, q)$.

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm{GL}_2$. Here, we use "hyperendoscopy" techniques to…

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…

Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of…

Let $P$ be a compact hyperbolic Coxeter truncation polytope of dimension $d\ge 3$, and let $\Gamma$ be the orbifold fundamental group of the associated Coxeter orbifold $\mathcal{O}_P$. Let $\mathscr{G}(\Gamma,G)$ be the geometric component…

We study a new lifting of automorphic representations using the theta representation $\Theta$ on the $4$-fold cover of the symplectic group, $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. This lifting produces the first examples of CAP…

In this memoir, we study the even unimodular lattices of rank at most 24, as well as a related collection of automorphic forms of the orthogonal, symplectic and linear groups of small rank. Our guide is the question of determining the…

I give a new integral representation for the degree five (standard) L-function for automorphic representations of GSp(4) that is a refinement of integral representation of Piatetski-Shapiro and Rallis. The new integral representation…

Based on Furusawa's theory, we present an integral representation for the L-function L(s,\pi \times \tau), where \pi is a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where \tau is an…

The first part of this paper studied $\mathrm{GSp}_4$-type abelian varieties and the corresponding compatible systems of $\mathrm{GSp}_4$ representations. Techniques in \cite{BCGP} are applied to show that one can prove the potential…

Let pi be a cuspidal, automorphic representation of GSp(4) attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,pi x tau), where tau runs…

Let $\pi$ be a polarized, regular algebraic, cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$ where $F$ is totally real or imaginary CM, and let $(\rho_\lambda)_\lambda$ be its associated compatible system of…

We show that a compact representation of a semisimple Lie group has an orthogonal decomposition into finite length representations. This generalises and simplifies a number of more special spectral theorems in the literature. We apply it to…

Let $\pi$ be a cohomological automorphic representation of $PGL(2)$ over a number field of arbitrary signature and assume that the local component of $\pi$ at a prime $\mathfrak{p}$ is the Steinberg representation. In this situation one can…

Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $\pi$ of $G$ over…

Let $F$ be a non-archimedean local field of characteristic zero. If $F$ has even residual characteristic, we assume $F/\mathbb{Q}_2$ is unramified. Let $V$ be a depth zero, irreducible, nongeneric supercuspidal representation of $GSp(4,…

In our previous paper we gave proofs of Ibukiyama's correspondences on holomorphic Siegel modular forms of degree 2 of half-integral weight and integral weight. In this paper, we formulate and prove similar correspondences on automorphic…

Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of…