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...

Behind this sophisticated title hides an elementary exercise on Clifford theory for index two subgroups and self-dual or conjugate-dual representations. When applied to semi-simple representations of the Weil–Deligne group W F ′ a non Archimedean local field F, further translated in terms GL n (F) via Langlands correspondence when has characteristic zero, it yields various statements concerning...

• The theory of admissible representations of GL(2,Qp) (or more generally, GL(2, F ) with F/Qp a finite extension). • The theory of automorphic representations of GL(2); in particular, the correspondence between Hecke eigenforms in the classical sense and automorphic representations. • The Jacquet-Langlands correspondence, relating automorphic forms on GL(2) with those on a division algebra. • ...

These notes were originally written in Fall 2010 to provide a very quick overview of some basic topics in modular forms, automorphic forms and automorphic representations. I have not made any significant changes since, or even proofread them completely (so some information may be outdated, and errors may remain), mostly just corrected some typos. If you spy any more errors, or have suggestions,...

A strong impetus for this paper came, at least for the first author, from a question of Avner Ash, asking whether one can construct non-self-dual, nonmonomial cuspidal cohomology classes for suitable congruence subgroups Γ of SL(n,Z), say for n = 6. Such a construction, in special examples, has been known for some time for n = 3 (see [2, 34, 35, 36]); it is of course not possible for n = 2. One...

In the late 1960s Robert Langlands launched what has become known as the Langlands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis [L]. In particular, Langlands conjectured that Galois representations and motives can be described in terms of the more tangible data of automorphic representations. A striking application of this general principle is...

Let π be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GLn(AF ), where F is a totally real field and n is at most 5. We show that for all primes l, the l-adic Galois representations associated to π are irreducible, and for all but finitely many primes l, the mod l Galois representations associated to π are also irreducible. We also show that the Lie algebras...

We consider periods of automorphic representations of adele groups defined by integrals along Gelfand subgroups. We define natural maps between local components of such periods and construct corresponding global maps using automorphic L-functions. This leads to an introduction of a global invariant of an automorphic representation arising from two such periods. We compute this invariant in some...

The almost automorphic solution is a generalization of the almost periodic solution. In this paper, the almost automorphic solutions of Cohen-Grossberg neural networks with delays are considered. Using the semi-discretization method and the contraction mapping principle, some sufficient conditions are obtained to ensure the existence and the uniqueness of almost automorphic solutions to Cohen-G...

(Under the Riemann Hypothesis the stronger bound |ζ(1 + it)| ≥ c log log t holds.) From a modern point of view, the method of de la Vallée Poussin is based on Rankin-Selberg convolutions and a positivity argument (an effective version of Landau’s Lemma – see [HL94, Appendix]). It can be applied to any Rankin-Selberg L-function L(s, π1 ⊗ π2), provided that one of the πi’s is self-dual (cf. [Sar0...