Local convolution of l-adic sheaves on the torus

Abstract

For K and L two l-adic perverse sheaves on the one-dimensional torus Gm,k¯ over the algebraic closure of a finite field, we show that the local monodromies of their convolution K ∗ L at its points of non-smoothness is completely determined by the local monodromies of K and L. We define local convolution bi-exact functors ρ(u) (s,t) for every s, t, u ∈ P 1 k¯that map continuous l-adic representations of the inertia groups at s and t to a representation of the inertia group at u, and show that the local monodromy of K ∗ L at u is the direct sum of the ρ(u) (s,t) applied to the local monodromies of K and L. This generalizes a previous result of N. Katz for the case where K and L are smooth, tame at 0 and totally wild at infinity.