We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct $p$-adic $L$-functions for non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the $L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the $p$-adic $L$-function of the eigenform to be this distribution.

Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\text{GS}{{\text{p}}_{4}}$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\text{PG}{{\text{L}}_{4}}$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\text{S}{{\text{L}}_{\text{4}}}\left( \text{C} \right)$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\text{PGU}\left( 2,\,2 \right)$ representing the product of the degree $8$ standard $L$-function and the degree $6$ exterior square $L$-function.

We prove a sharp upper bound on the L2 norm of a GL3 Maass form restricted to GL2×ℝ+.

Let K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

we study some automorphic cohomology classes of degree one on the griffiths–schmid varieties attached to some unitary groups in three variables. using partial compactifications of those varieties, constructed by kato and usui, we define for such a cohomology class some analogues of fourier–shimura coefficients, which are cohomology classes on certain elliptic curves. we show that a large space of such automorphic classes can be generated by those with rational ‘coefficients’. more precisely, we consider those cohomology classes that come from picard modular forms, via some penrose-like transform studied in a previous article: we prove that the coefficients of the classes thus obtained can be computed from the coefficients of the picard form by a similar transform defined at the level of the elliptic curve.

The local intersection cohomology of a point in the Baily–Borel compactification (of a Hermitian locally symmetric space) is shown to be canonically isomorphic to the weighted cohomology of a certain linear locally symmetric space (an arithmetic quotient of the associated self-adjoint homogeneous cone). Explicit computations are given for the symplectic group in four variables.

Our aim is to study the cohomology groups of some coherent sheaves on a Griffiths–Schmid variety associated with an anisotropic Q-from of the unitary group SU(2,1). We define some transforms relating this cohomology to the coherent cohomology groups of some sheaves defined on certain threefolds, which are fibered in projective lines over Picard modular surfaces. In particular, we give a complete and explicit description, in terms of classical Picard modular forms, of the holomorphic (resp. anti-holomorphic) part of the 1-cohomology of the Griffiths-Schmid variety. From this, it results an explicit generating system for the part of the 2–cohomology which correspond to those automorphic representations whose archimedean component is a degenerate limit of discrete series.

Let $G$ be an inner anisotropic form of an unitary group of 3 variables over Q, such that $G_R≃U(2,1)$, and π be an automorphic representation of G(A) whose archimedean component π∞ is a degenerate limit of discrete series; such a π never occurs in the cohomology (coherent or étale) of a Shimura variety. We show that however it does ‘appear’ in the coherent cohomology of some line bundle over an associated Griffiths-Schmid variety. Moreover we study cup products between such cohomology classes and some other automorphic cohomology classes and we prove some non-vanishing results.