Sums of Hecke eigenvalues along polynomial sequences and base change for $\text{GL}(2)$

TL;DR

This paper investigates the behavior of sums of Hecke eigenvalues for GL(2) representations, establishing conditions for logarithmic savings and linking these sums to base change problems.

Contribution

It provides new criteria for when sums of Hecke eigenvalues exhibit savings and connects these sums to the base change problem for GL(2).

Findings

Sums of Hecke eigenvalues show logarithmic savings only for cuspidal representations.

The problem of summing Hecke eigenvalues along polynomial sequences relates to GL(2) base change.

The study characterizes when these sums outperform trivial bounds.

Abstract

We study sums of absolute values of Hecke eigenvalues of $\textrm{GL}(2)$ representations that are tempered at all finite places. We show that these sums exhibit logarithmic savings over the trivial bound if and only if the representation is cuspidal. Further, we connect the problem of studying the sums of Hecke eigenvalues along polynomial values to the base change problem for $\textrm{GL}(2).$