Algebraic twists of modular forms and Hecke orbits

TL;DR

This paper proves that Fourier coefficients of modular forms are uncorrelated with algebraic functions, demonstrating equidistribution of twisted Hecke orbits using advanced methods like the amplification technique and the Riemann Hypothesis over finite fields.

Contribution

It establishes the absence of correlation between modular form Fourier coefficients and algebraic functions, extending understanding of Hecke orbit equidistribution.

Findings

Proves uncorrelation with algebraic functions with power savings.

Demonstrates equidistribution of twisted Hecke orbits.

Utilizes the amplification method and Deligne's Fourier transform.

Abstract

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the ell-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.