Distinguishing Galois representations by their normalized traces

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.