Distribution of Hecke eigenvalues for automorphic representations

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My program is focused on the distribution of Hecke eigenvalues for automorphic representations via the study of automorphic L-functions. The study of L-functions has a rich history, going back to the Riemann zeta function, where analytic properties of the function were shown to correspond to arithmetic information (for example, the divergence of the Euler product for the Riemann zeta function at 1 implies the infinitude of primes). My particular interest lies in the connection between the Langlands functoriality conjectures and distribution results for Hecke eigenvalues. A related example is Serre's work in the setting of Galois representations, where he showed that the Sato-Tate conjecture is implied by certain analytic properties of symmetric power L-functions. My program aims to develop new results on the distribution of Hecke eigenvalues from two different perspectives. One aspect of the program considers the following question: Given two distinct cuspidal automorphic representations for GL(n) over a number field, what can be said about the size of the set S of finite unramified places at which their associated Hecke eigenvalues differ? A classical result of Jacquet-Shalika gave a response to this question by showing that S was infinite, which is known as the strong multiplicity one theorem. Further progress on such questions took place through the work of Ramakrishnan, Murty-Rajan, Rajan, and others. I plan to fill out this picture by demonstrating how the incremental use of functoriality results translates into progressively stronger statements about the size of the set S. The aim is to quantify the rapport between conjectures within the Langlands program and refinements of strong multiplicity one. Another aspect consists of studying the sequence of Hecke eigenvalues associated to a single automorphic representation. We fix a constant and ask how often a cuspidal automorphic representation has a Hecke eigenvalue equal to that constant. Analogous questions have been raised by Serre and Lang-Trotter in the setting of Galois representations. In earlier work, I provided an answer in the GL(2) setting to the question of Serre by applying results on the automorphy of symmetric powers and obtaining upper bounds on the occurrence of a fixed constant as a Hecke eigenvalue. I aim to improve on the above strategy and strengthen the bounds. Another consequence of incorporating this improvement is that it should be amenable to the application of large symmetric powers and therefore enable me to establish a succession of bounds based on incremental assumptions about the conjectured automorphy of the symmetric power lifts. I also plan to obtain unconditional results about the occurrence of Hecke eigenvalues in different regions of the complex plane, building on techniques from earlier work of mine and others.