Distinguishing Galois representations by their normalized traces

Vijay M. Patankar; C. S. Rajan

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arXiv · 1609.09724

Distinguishing Galois representations by their normalized traces

Vijay M. Patankar·C. S. Rajan

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Abstract

Suppose $ ρ_1 $ and $ ρ_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(ρ_1(σ_v)) /Nv^{k_1/2}$ and $ Tr(ρ_2(σ_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 $ρ_1$ is connected and the repesentation is absolutely irreducible. We prove that $ ρ_1 $ and $ ρ_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.

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BibTeX RIS

Vijay M. Patankar, C. S. Rajan. 2016-09-30. Distinguishing Galois representations by their normalized traces. https://arxiv.org/abs/1609.09724

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