Distinguishing pure representations by normalized traces

Abstract

Given two pure representations of the absolute Galois group of an $\ell$-adic number field with coefficients in ${\overline{\mathbb{Q}}}_p$ (with $\ell \ne p$), we show that the Frobenius-semisimplifications of the associated Weil--Deligne representations are twists of each other by an integral power of certain unramified character if they have equal normalized traces. This is an analogue of a recent result of Patankar and Rajan in the context of local Galois representations.