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Let F be a global field and A its ring of adeles. Let G:=SL(2). We study the bilinear form B on the space of K-finite smooth compactly supported functions on G(A )/G(F) defined by the formula B (f,g):=B(f,g)-(M^{-1}CT (f),CT (g)), where B is the usual scalar product, CT is the constant term operator, and M is the standard intertwiner. This form is natural from the viewpoint of the geometric Langlands program. To justify this claim, we provide a dictionary between the classical and geometric theory of automorphic forms. We also show that the form B is related to S. Schieders Picard-Lefschetz oscillators.
We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also
We consider a general class of Fourier coefficients for an automorphic form on a finite cover of a reductive adelic group ${bf G}(mathbb{A}_{mathbb{K}})$, associated to the data of a `Whittaker pair. We describe a quasi-order on Fourier coefficients,
We give an asymptotic formula for the number of automorphic forms on the non-split norm one torus $T$ associated with an imaginary quadratic extension of $mathbb{Q}$, ordered by analytic conductor.
Conjecturally, the Galois representations that are attached to essentially selfdual regular algebraic cuspidal automorphic representations are Zariski-dense in a polarized Galois deformation ring. We prove new results in this direction in the context
In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic $p$). We first introduce a bialgebra $mathcal{H}$ and we then introduce a notion of $mathcal{H}$-module vertex algebra and a noti