Course Identification

Good knowledge of Representation Theory.
Basic knowledge of the theory of algebraic groups.
Knowledge of Complex Analysis.
Some knowledge of representation theory of real and p-adic reductive
groups will be quite useful.

1. Generalities on Automorphic Representations. Global fields - Arithmetic and Geometric cases.
2. Theory of Eisenstein series for automorphic representations.
3. Meromorphic continuation and Functional Equation for Eisenstein Series.
4. Langlands L-functions associated to automorphic representations. Langlands conjectures.
5. Functoriality conjecture and Base change lifting.
6. Automorphic periods. Product formulas. Mutiplicity one results (local and global).
7. Relations between Automorphic periods and L-functions.
8. General theory of L-functions. General Riemann Conjecture and Lindelof conjecture.
9. Convexity and subconvexity bounds for L-functions and Automorphic Periods.
10.  Introduction to Trace Formula and Relative Trace Formula for Automorphic Representations.

Upon successful completion of this course, the students will be able to:

1. Demonstate knowlege of the deep language of L-functions and automorphic representations.
2. Gain access to the modern literature on analytic number theory.
3. Introduce the L-functions related to automorphic representations.
4. Get information about automorphic representations using automorphic periods.
5. Relate L-functions with Automorphic periods. 

Book "Introduction to Langlands program" by J. Bernstein , S. Gelbart, S.S. Kudla, E. Kowalski, E. de Shalit, D. Gaitsgory, J.W. Cogdell, D. Bump.
Book " Automorphic Forms and Representations of Adele Groups" by S. Gelbart
Paper "Perspectives of the Analytic Theory of L-functions" by H. Iwaniec and P. Sarnak