Local convolution of l-adic sheaves on the torus

TL;DR

This paper proves that the local monodromies of the convolution of two l-adic perverse sheaves on the torus are fully determined by their individual local monodromies, extending previous results to more general cases.

Contribution

It generalizes Katz's earlier work by establishing the behavior of local monodromies for convolutions of possibly wild sheaves on the torus.

Findings

Local monodromies of convolution are determined by individual monodromies.

Extension of Katz's results to wild sheaves.

Provides a comprehensive description of monodromy behavior at non-smooth points.

Abstract

For K and L two l-adic perverse sheaves on the one-dimensional torus 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. 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.