Algebraic trace functions over the primes

TL;DR

This paper develops new bounds for sums over primes involving trace functions of $\,ell$-adic sheaves, extending previous work to Eisenstein series and applying finite field Riemann Hypothesis techniques.

Contribution

It introduces general power-saving estimates for prime sums of trace functions and extends algebraic twist results to Eisenstein series.

Findings

Established bounds with power-saving for sums over primes of trace functions.

Extended algebraic twist results to Eisenstein series.

Provided concrete applications of the theoretical estimates.

Abstract

We study sums over primes of trace functions of $\ell$-adic sheaves. Using an extension of our earlier results on algebraic twist of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann Hypothesis over finite fields, we prove general estimates with power-saving for such sums. We then derive various concrete applications.