Let X be a smooth projective curve. Write BunSO2n for the moduli stack of SO2n-torsors on X. We give a geometric interpretation of the automorphic function f on BunSO2n corresponding to the minimal representation. Namely, we construct a perverse sheaf KH on BunSO2n such that f should be equal to the trace of the Frobenius of KH plus some constant function. The construction is based on some expl...

In the first part of this paper we study minimal representations of simply connected simple split groups G of type Dk or Ek over local nonarchimedian fields. Our main result is an explicit formula for the spherical vectors in these representations. In the case of groups over R and C, such a formula was obtained recently in [8]. We also use our techniques to study the structure of the space of s...

We prove a new upper bound for the dimension of space cohomological automorphic forms fixed level and growing parallel weight on $\mathrm{GL}_2$ over number field which is not totally real, improving one obtained by Marshall. The main tool proof mod $p$ representation theory $\mathrm{GL}_2(\mathbb{Q}_p)$ as started Barthel-Livne Breuil, developed Paskunas.

We study discrete almost automorphic functions sequences defined on the set of integers with values in a Banach space X. Given a bounded linear operator T defined on X and a discrete almost automorphic function f n , we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation Δu n Tu n f n . We also prove the existence of a discrete almost autom...

Often p-adic L-functions are the right bridge between these and the arithmetic objects. p-adic L-functions: • These admit no näıve definition. • In general, given M a “motive”or “algebraic automorphic representation”, a p-adic L-function associated with M tends to have two properties: – it interpolates the special values L(n,M ⊗ χ) p-adically, for n in some interval and χa p-power conductor Dir...

Let p be a prime and F totally real field in which is unramified. We consider mod Hilbert modular forms for F, defined as sections of automorphic line bundles on varieties level to characteristic p. For Hecke eigenform arbitrary weight (without parity hypotheses), we associate two-dimensional representation the absolute Galois group give conjectural description set weights all eigenforms from i...

The following are extended notes of a lecture given by the author at the international colloquium on L-functions and Automorphic Representation held at TIFR in january 2012. This lecture reported on some joint work of Chris Skinner and the author on the link between central L-values and Selmer groups of elliptic curves. The detailed proofs of our results will appear in [SU13]. The author presen...

Drinfeld’s relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting of stable maps from nodal deformations of the curve into twisted flag varieties. As an application, we prove that the twisted intersection coho...

We study the L-functions associated to Siegel modular forms — equivalently, automorphic representations of GSp(4, ‫ށ‬ ‫ޑ‬) — both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform representation theoretic calculations to cast the Langlands L-function in classical terms. We develop a precise notion of what it means to test a conjectured functional equation f...

The classical theory of Gelfand pairs has found a wide range of applications, ranging from harmonic analysis on Riemannian symmetric spaces to coding theory. Here we discuss a generalization of this theory, due to Gelfand-Kazhdan, and Bernstein, which was developed to study the representation theory of p-adic groups. We also present some recent number-theoretic results, on local e-factors and o...