Comparing Hecke Coefficients of Automorphic Representations

TL;DR

This paper establishes statistical bounds and distributional results for Hecke coefficients of automorphic representations over number fields, extending to higher rank groups under functoriality assumptions.

Contribution

It provides unconditional bounds on Hecke coefficients, extends methods to GL(n) assuming functoriality, and addresses questions about large eigenvalues of Maass forms.

Findings

Bounds on places with negative linear combinations of Hecke coefficients

Distributional results of Hecke coefficients under Ramanujan conjecture

Extension of methods to GL(n) assuming functoriality

Abstract

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of $\operatorname{GL}(2)$ over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin-Selberg $L$-functions, we obtain bounds on the set of places where linear combinations of Hecke coefficients are negative. Under a mild functoriality assumption we extend these methods to $\operatorname{GL}(n)$. As an application, we obtain a result related to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan conjecture is satisfied, we obtain distributional results of the Hecke coefficients at places varying in certain congruence or Galois classes.