Algebraic twists of automorphic L-functions

Abstract: Let $(\lambda(n))_{n\geq 1}$ denote the sequence of coefficients of an automorphic $L$-function. Let $q$ be square free and $K:Z/qZ\mapsto C$ be a $q$-periodic function build from « trace functions «  from $\ell$-adic sheaves on affine lines in characteristics dividing $q$.
 
In this talk, based on joint works with E. Fouvry, E. Kowalski, Y. Lin, C. Raju and W. Sawin, we will discuss various results and techniques concerning the evaluation of sums of the shape
$$\sum_{n\leq X}\lambda(n)K(n)$$
as $q,X$ tend to $\infty$ at suitable relative speeds.

This generalisation from the traditional case of twists by multiplicative characters is useful when investigating, for instance, the distribution of the $\lambda(n)$ in long arithmetic progression modulo $q$.

This talk is dedicated to the memory of Chandrasekhar Raju.