On the Matching Density of Icosahedral and Tetrahedral Representations

Abstract

The matching density of two Galois Representations ρ and η is the Dirichlet density of the set of finite places where ρ and η are unramified and the traces of Frobenius of ρ and η are equal. Using matching density, we can quantify the similarity of two Galois representations. In this paper, we find sharp upper bounds for the matching density of certain pairs of tetrahedral and icosahedral representations.