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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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Computing Vf modulo p

Computing Vf modulo p

(p.337) Chapter Thirteen Computing Vf modulo p
Computational Aspects of Modular Forms and Galois Representations

Jean-Marc Couveignes

Princeton University Press

This chapter addresses the problem of computing in the group of lsuperscript k-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing modular representations. An algorithm in this chapter usually means a probabilistic Las Vegas algorithm. In some places it gives deterministic or probabilistic Monte Carlo algorithms, but this will be stated explicitly. The main reason for using probabilistic Turing machines is that there is a need to construct generating sets for the Picard group of curves over finite fields. Solving such a problem in the deterministic world is out of reach at this time. The unique goal is to prove, as quickly as possible, that the problems studied in this chapter can be solved in probabilistic polynomial time.

Keywords:   modular forms, plane curves, random divisors, modular representations, Las Vegas algorithm, Turing machines, probabilistic polynomial time

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