Local Convolution of `-adic Sheaves on the Torus

For K and L two `-adic perverse sheaves on the one-dimensional torus Gm,k̄ 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 ∈ P1 k̄ that map continuous `-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.