‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...

• 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. • ...

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We ...

• 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.

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 I have not made any significant changes since, or even proofread them completely (so some information may be outdated, and errors may remain), mostly just corrected some typos.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 prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hi...

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...

Part I. The origins of the Langlands Program 9 1. The Langlands correspondence over number fields 9 1.1. Galois group 9 1.2. Abelian class field theory 10 1.3. Frobenius automorphisms 13 1.4. Rigidifying ACFT 14 1.5. Non-abelian generalization? 15 1.6. Automorphic representations of GL2(AQ) and modular forms 18 1.7. Elliptic curves and Galois representations 22 2. From number fields to function...