Trace formulas and their applications on Hecke eigenvalues

Abstract

The objective of the thesis is to investigate the trace formulas and their applications on Hecke eigenvalues, especially on the distribution of Hecke eigenvalues. This thesis is divided into two parts..

In the first part of the thesis, a review is firstly carried out for the equidistribution of Hecke eigenvalues as primes vary and for the expected size of the error term in this equidistribution problem. Then the Kuznetsov trace formula is applied to prove a result on the size of the error term in the asymptotic distribution formula of Hecke eigenvalues. These eigenvalues become equidistributed with respect to the p-adic Plancherel measures as Hecke eigenforms vary. Next, this problem is generalized to Satake parameters of GL2 representations with prescribed supercuspidal local representations. Such a generalization is novel to the case of classical automorphic forms. To achieve this result, a trace formula of Arthur-Selberg type with a couple of key refinements is used.

In the second part of the thesis, a density theorem is proved which counts the number of exceptional nontrivial zeros of a family of symmetric power L-functions attached to primitive Maass forms in the critical strip. In addition, a large sieve inequality of Elliott-Montgomery-Vaughan type for primitive Maass forms is established. The density theorem and large sieve inequality have many applications. For instance, they are used to prove statistical results on Hecke eigenvalues of primitive Maass forms and the extreme values of the symmetric power L-functions attached to primitive Maass forms.