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

Abstract

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