Matching densities for Galois representations

TL;DR

This paper introduces the concept of matching density for pairs of Galois representations over Q, showing that their possible densities are densely distributed in the interval [0, 1], with implications for automorphic representations under certain conjectures.

Contribution

It establishes that the set of matching densities for pairs of irreducible Galois representations is dense in [0, 1], extending to automorphic representations under the strong Artin conjecture.

Findings

Matching densities are dense in [0, 1] for pairs of irreducible Galois representations.

Implication of results for automorphic representations assuming the strong Artin conjecture.

Provides a new perspective on the distribution of Frobenius trace coincidences.

Abstract

Given a pair of n-dimensional complex Galois representations over Q, we define their matching density to be the density, if it exists, of the set of places at which the traces of Frobenius of the two Galois representations are equal. We will show that the set of matching densities of such pairs of irreducible Galois representations (for all n) is dense in the interval [0, 1]. Under the strong Artin conjecture, this also implies the corresponding statement for cuspidal automorphic representations.