The cohomology of an arithmetic group is built out of certain automorphic forms. This allows computational investigation of these automorphic forms using topological techniques. We discuss recent techniques developed for the explicit computation of the cohomology of congruence subgroups of GL2 over CM-quartic and complex cubic number fields as Hecke-modules.

For a cuspidal automorphic representation Π of GL(4, A), H. Kim proved that the exterior square transfer ∧Π is nearly an isobaric automorphic representation of GL(6, A). In this paper we characterize those representations Π for which ∧Π is cuspidal.

In this article we define the concept of weighted pseudo almost automorphic sequence, and establish some basic properties of these sequences. Further, as an application, we show the existence, uniqueness and global attractivity of weighted pseudo almost automorphic sequence solutions of a neural network model.

We describe the structure of a free loop of rank n in the variety of automorphic Moufang loops as a subdirect product of a free group and a free commutative Moufang loop, both of rank n. In particular, the variety of automorphic Moufang loops is the join of the variety of groups and the variety of commutative Moufang loops.

We discuss a set of heterotic and type II string theory compactifications to 1+1 dimensions that are characterized by factorized internal worldsheet CFTs the form $V_1\otimes \bar V_2$, where $V_1, V_2$ self-dual (super) vertex operator algebras. In cases with spacetime supersymmetry, we show BPS states module for Borcherds-Kac-Moody (BKM) (super)algebra, prove each model BKM (super)algebra is ...

We study automorphic Lie algebras and their applications to integrable systems. Automorphic are a natural generalisation of celebrated Kac-Moody the case when group automorphisms is not cyclic. They infinite dimensional almost graded. formulate concept graded isomorphism classify s l ( 2 , C ) based corresponding all finite reduction groups. show that hierarchies systems, Lax representations ma...

It is proved that certain types of modular cusp forms generate irreducible automorphic representations of the underlying algebraic group. Analogous Archimedean and non-Archimedean local statements are also given. Introduction One of the motivations for this paper was to show that full level cuspidal Siegel eigenforms generate irreducible, automorphic representations of the adelic symplectic sim...

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry group G. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundl...

The Langlands philosophy contemplates the relation between auto-morphic representations and Galois representations. A particularly interesting case is that of the non-selfdual automorphic representations of GL 3. Clozel conjectured that the L-functions of certain of these are equal to L-functions of Galois representations. Here we announce that we found an example of such an automorphic represe...

Ash's research was partially supported by NSF Grant DMS-8919696. Conjecturally, any “algebraic” automorphic representation on GL(n) should have an n-dimensional Galois representation attached. Many examples of algebraic automorphic representations come from the cohomology overC of congruence subgroups of GL(n;Z). On the other hand, the first author has conjectured that for any Hecke eigenclass ...