This proposal is concerned with the analytic theory of automorphic forms. Specifically the study of the size of a general automorphic L-function on the critical line and the size of an eigenfunction on an arithmetic locally symmetric space .In both cases the main goal is to establish a subconvex estimate . In the first case the sharpest form of such an estimate would follow from the Grand Riemann Hypothesis .However many of the desired applications of such estimates only require subconvexity. The applications are varied ,one such being the recent subconvex estimate by Cogdell,Piatetsky-Shapiro and the proposer which allows for the resolution of Hilbert's 11th problem on the representation of integers by quadratic forms in a number field.Other applications are to problems in mathematical physics ,specifically the behavior of quantum states in quantizations of classically chaotic systems. The second problem of the size of an eigenfunction on such locally symmetric spaces is closely associated with the first and the proposal is concerned with understanding this more difficult problem.It constitutes a generalization of the Ramanujan conjectures.
(general): The proposal is concerned with the study of special types of geometric spaces which are defined via arithmetic and number theoretic constructions. This has been an active area of investigation for the last 60 or so years ,primarily since it carries some of the most powerful tools that we know in number theory (to diophantine problems as well as ones associated with prime numbers). Specifically the seemingly technical issues that are being investigated have applications to resolve some simple well known problems in number theory. One such is one of Hilbert's problems from 1900 concerning which numbers are sums of 3 integer squares in an extention of the ordinary whole numbers to integers in a number field. Other applications of this theory are perhaps less traditional and are to Mathematical Physics,specifically Quantum Chaos. These spaces provide one of the few (in fact the only one) classically chaotic Hamiltonian systems whose quantization can be satisfactorally studied mathematically. The techniques to do so are number theoretic and the results are often quite surprizing.
