Tensor and convolution direct image of $\ell$-adic sheaves

TL;DR

This paper introduces tensor and convolution direct image operations for $\, ext{ell}$-adic sheaves under Galois étale maps, enhancing estimates on exponential sums and rational points on varieties.

Contribution

It defines and studies tensor and convolution direct image functors for $\, ext{ell}$-adic sheaves, providing new tools for arithmetic geometry applications.

Findings

Improved bounds on partial exponential sums.

Enhanced estimates on the number of rational points.

Development of new cohomological operations for $\, ext{ell}$-adic sheaves.

Abstract

Given a Galois \'etale map of varieties $\pi:Y\to X$ and an $\ell$-adic sheaf or derived category object $P\in D^b_c(Y,{\mathbb Q}_\ell)$, we study two cohomological operations: the tensor direct image and (in the case of perverse sheaves) the convolution direct image, which give objects of $D^b_c(X,{\mathbb Q}_\ell)$, and can be used to improve the estimates on some partial exponential sums and the number of rational points on certain varieties.