Distinguishing Galois representations by their normalized traces

Abstract

Suppose ρ1 and ρ2 are two pure ℓ-adic degree n representations of the absolute Galois group of a number field K of weights k1 and k2 respectively, having equal normalized Frobenius traces Tr(ρ1(σv))/Nvk1/2 and Tr(ρ2(σv))/Nvk2/2 at a set of primes v of K with positive upper density. Assume further that the algebraic monodromy group of ρ1 is connected and ρ1 is absolutely irreducible. We prove that ρ1 and ρ2 are twists of each other by a power of the ℓ-adic cyclotomic character times a character of finite order. As a corollary, we deduce a theorem of Murty and Pujahari proving a refinement of the strong multiplicity one theorem for normalized eigenvalues of newforms.