Distinguishing Galois representations by their normalized traces

TL;DR

This paper proves that two pure Galois representations with equal normalized Frobenius traces at a positive density set of primes are essentially twists of each other, under certain conditions, with applications to modular forms.

Contribution

It establishes a criterion for distinguishing Galois representations based on normalized traces, extending previous results and applying to modular forms.

Findings

Two Galois representations with equal normalized traces are twists of each other.

The result applies to modular forms, confirming a known theorem.

Conditions include connected algebraic monodromy and irreducibility.

Abstract

Suppose \( \rho_1 \) and \( \rho_2 \) are two pure Galois representations of the absolute Galois group of a number field $K$ of weights \( k_1 \) and \( k_2 \) respectively, having equal normalized Frobenius traces \( Tr(\rho_1(\sigma_v)) /Nv^{k_1/2}\) and \( Tr(\rho_2(\sigma_v)) /Nv^{k_2/2}\) at a set of primes \( v\) of $K$ with positive upper density. Assume further that the algebraic monodromy group of $\rho_1$ is connected and the repesentation is absolutely irreducible. We prove that \( \rho_1 \) and \( \rho_2 \) are twists of each other by power of a Tate twist times a character of finite order. We apply this to modular forms and deduce a result proved by Murty and Pujahari.