We construct a Langlands parameterization of supercuspidal representations $G_2$ over $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set generic $G_2(K)$ to irreducible continuous homomorphisms $\rho W_K \to G_2(\CC)$ with $W_K$ Weil group $K$. The construction map is simply matter assembling ...

• 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 I have not made any significant changes since, or even proofread them completely (so some information may be outdated, and errors may remain), mostly just corrected some typos.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,...

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 theta correspondence 1 is a correspondence between automorphic forms on two members of a dual reductive pair. Because of the relation between automorphic forms and automor-phic representations, it is also a correspondence between automorphic representations. The Siegel-Weil formula says that certain natural linear combinations of theta lifts are Siegel-type holomorphic Eisenstein Series.

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...

I give an algorithm for computing the full space of automorphic forms for definite unitary groups over Q, and apply this to calculate the automorphic forms of level G(Ẑ) and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U1 × U1 × U1 and U1 × U2, and to an example of a non-endoscopic f...

In the paper ‘Automorphic functions for a Whitehead-complement group’ [5], Matsumoto, Nishi and Yoshida constructed automorphic functions on real 3–dimensional hyperbolic space for a Kleinian group called the Whitehead-link-complement group. For a Kleinian group (of the first kind), no automorphic function/form has been studied before. In this note, their motivation is presented with a historic...

We are concerned with the almost automorphic solutions to the second-order elliptic differential equations of type ü(s) + 2Bu̇(s) + Au(s) = f(s) (∗), where A, B are densely defined closed linear operators acting in a Hilbert space H and f : R 7→ H is a vector-valued almost automorphic function. Using invariant subspaces, it will be shown that under appropriate assumptions; every solution to (∗) ...