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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 the residual Galois representations

Computing the residual Galois representations

(p.371) Chapter Fourteen Computing the residual Galois representations
Computational Aspects of Modular Forms and Galois Representations

Bas Edixhoven

Princeton University Press

This chapter proves the main result on the computation of Galois representations. It provides a detailed description of the algorithm and a rigorous proof of the complexity. It first combines the results of chapters 11 and 12 in order to work out the strategy of Chapter 3. This gives the main result, Theorem 14.1.1: a deterministic polynomial time algorithm, based on computations with complex numbers. The crucial transition from approximations to exact values is done, and the proof of Theorem 14.1.1 is finished later in the chapter. The chapter then replaces the complex computations with the computations over finite fields from Chapter 13, and gives a probabilistic (Las Vegas type) polynomial time variant of the algorithm in Theorem 14.1.1.

Keywords:   modular forms, Galois representations, polynomial time algorithm

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