Locally potentially equivalent Galois representations

2010/10/26 by Patankar, Vijay M., Rajan, C. S.
#FOS: Mathematics #Number Theory (math.NT) #Primary 11F80 #Secondary 11R45

paper · doi:10.48550/arxiv.1010.5393

Abstract

We show that if two continuous semi-simple \(ℓ \)-adic Galois representations are locally potentially equivalent at a sufficiently large set of places then they are globaly potentially equivalent. We also prove an analogous result for arbitrarily varying powers of character values evaluated at the Frobenius conjugacy classes. In the context of modular forms, we prove: given two non-CM newforms f and g of weight at least two, such that a_p(f)^n_p=a_p(g)^n_p on a set of primes of positive upper density and for some set of natural numbers n_p, then f and g are twists of each other by a Dirichlet character.

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