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Computational Aspects of Modular Forms and Galois RepresentationsHow One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)$
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Bas Edixhoven and Jean-Marc Couveignes

Print publication date: 2011

Print ISBN-13: 9780691142012

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691142012.001.0001

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Polynomials for projective representations of level one forms

Polynomials for projective representations of level one forms

(p.159) Chapter Seven Polynomials for projective representations of level one forms
Computational Aspects of Modular Forms and Galois Representations

Johan Bosman

Princeton University Press

This chapter explicitly computes mod-ℓ Galois representations attached to modular forms. To be precise, it looks at cases with l ≤ 23, and the modular forms considered will be cusp forms of level 1 and weight up to 22. The chapter presents the result in terms of polynomials associated with the projectivized representations. As an application, it will improve a known result on Lehmer's nonvanishing conjecture for Ramanujan's tau function.

Keywords:   modular forms, Galois representations, cusp forms, polynomials, Ramanujan's tau function, Lehmer, nonvanishing conjecture

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