Algebraic twists of modular forms and Hecke orbits

Étienne Fouvry, Emmanuel Kowalski, Philippe Michel

Published 2012-07-03, updated 2014-11-16Version 5

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.