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

Given a Galois 'etale map of varieties $\pi :Y\to X$ and an $\mathrm{\ell }$-adic sheaf or derived category object $P\in D_c^b(Y,{\mathbb{Q}}_{\mathrm{\ell }})$, 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.