Mathematical Sciences: Automorphic Forms and Representation Theory

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This research deals with the subject of automorphic forms and representation theory using geometric methods. The main problem is to obtain the analytic continuation and functional equation of the Hasse-Weil zeta function of Shimura varieties. The object is to express these zeta functions in terms of automorphic L-functions. Then one must obtain the transformation laws for these automorphic L-functions. This research will study the generating functions for the number of solutions of systems of polynomial equations modulo all the primes. The properties of these functions will give an immense amount of information concerning the solution to the corresponding set of Diophantine Equations. Number Theory has as one of its chief components the study of Diophantine Equations, that is the solution of sets of equations in integers.