Distinguishing Galois representations by their normalized traces

2016/09/30 by Vijay M. Patankar, Patankar, Vijay M., C. S. Rajan +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1609.09724

openalex publication_date 2016/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose \( ρ₁ \) and \( ρ₂ \) are two pure Galois representations of the absolute Galois group of a number field K of weights \( k₁ \) and \( k₂ \) respectively, having equal normalized Frobenius traces \( Tr(ρ₁(σ_v)) /Nv^k₁/2\) and \( Tr(ρ₂(σ_v)) /Nv^k₂/2\) at a set of primes \( v\) of K with positive upper density. Assume further that the algebraic monodromy group of ρ₁ is connected and the repesentation is absolutely irreducible. We prove that \( ρ₁ \) and \( ρ₂ \) 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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