P-adic L-functions in universal deformation families

We construct examples of p-adic L-functions over universal deformation spaces for $${{\,\mathrm{GL}\,}}_2$$ GL 2 . We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions and Euler systems are not the usual eigenvarieties (parametrising nearly-ordinary families of automorphic representations), but other, larger spaces depending on a choice of a parabolic subgroup, which we call ‘big parabolic eigenvarieties’.

An Euler system for GU(2, 1)

We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of $${{\,\mathrm{GL}\,}}_3$$ GL 3 over imaginary quadratic fields, using the cohomology of Shimura varieties for $${\text {GU}}(2, 1)$$ GU ( 2 , 1 ) .

A geometric Jacquet–Langlands correspondence for paramodular Siegel threefolds

We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. We give some applications to the Langlands programme: using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet–Langlands correspondence between $${\text {GSp}}_4$$ GSp 4 and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $${\text {GSp}}_4$$ GSp 4 .

On counting cuspidal automorphic representations for GSp(4)

We find the number s k ⁢ ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ⁢ ( 4 , A Q ) \mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ⁢ ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

HECKE OPERATORS AND THE COHERENT COHOMOLOGY OF SHIMURA VARIETIES

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

RAISING THE LEVEL OF AUTOMORPHIC REPRESENTATIONS OF OF UNITARY TYPE

We use the endoscopic classification of automorphic representations of even-dimensional unitary groups to construct level-raising congruences.

A conjectural refinement of strong multiplicity one for GL(n)

Given a pair of distinct unitary cuspidal automorphic representations for GL([Formula: see text]) over a number field, let [Formula: see text] denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this paper, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin–Selberg products imply lower bounds on the size of [Formula: see text]. We also obtain further results for GL(3).

Poles of triple product L-functions involving monomial representations

In this paper, we study the order of the pole of the triple tensor product [Formula: see text]-functions [Formula: see text] for cuspidal automorphic representations [Formula: see text] of [Formula: see text] in the setting where one of the [Formula: see text] is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands’ beyond endoscopy proposal.