# Computing the residual Galois representations

# Computing the residual Galois representations

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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