Abstract
Given two pure representations of the absolute Galois group of an ℓ-adic number field with coefficients in Q‾p (with ℓ≠p), we show that the Frobenius-semisimplifications of the associated Weil–Deligne representations are twists of each other by an integral power of a 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.
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