Number Theory Seminar

We obtain non-trivial bounds for bilinear sums involving the trace function of an ell-adic sheaf over a finite field Fq​, which go well below the Polya-Vinogradov range, under a simple structural assumption on its geometric monodromy group. This is in contrast to previous works which required rather precise informations on the sheaf (such as the shape of its local monodromies at specific points along with global informations). Our approach is « soft » and builds on the method of moments due to J. Xu, a robust version of the Goursat-Kolchin-Ribet criterion and on Sawin's quantitative sheaf theory. In this talk we will describe the proof along with some applications.This is joint work with Etienne Fouvry, Emmanuel Kowalski and Will Sawin: https://arxiv.org/abs/2511.09459