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Section: New Results

Complex multiplication and modularity

Participants : Jean-Marc Couveignes, Andreas Enge, Nicolas Mascot, Enea Milio, Aurel Page, Damien Robert.

A. Enge and E. Thomé describe in [20] a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, they pursue an approach due to Dupont for evaluating $\theta$-constants in quasi-linear time using Newton iterations on the Borchardt mean. They report on experiments with the implementation Cmh and present an example with class number 20016.

In [34] E. Milio explains how to generalise the work of Régis Dupont for computing modular polynomials in dimension 2 to invariants derived from theta constants. Modular polynomials have many applications. In particular, they could speed up the CRT-algorithm to compute class fields of degree 4 CM-fields which would lead to faster algorithms to construct cryptographically secure Jacobians of hyperelliptic curves. They are also used to compute graphs of isogenies. This paper presents how to compute modular polynomials and the polynomials computed and then it proves some of their properties.

With F. Morain, A. Enge has determined exhaustively under which conditions “generalised Weber functions”, that is, simple quotients of $\eta$ functions of not necessarily prime transformation level and not necessarily of genus 1, yield class invariants [19] . The result is a new infinite family of generators for ring class fields, usable to determine complex multiplication curves. They examine in detail which lower powers of the functions are applicable, thus saving a factor of up to 12 in the size of the class polynomials, and describe the cases in which the polynomials have integral rational instead of integral quadratic coefficients.

N. Mascot has continued his work on computing Galois representations attached to Jacobians of modular curves. He has given tables of modular Galois representations in [33] obtained using the algorithm of [39] . He has computed Galois representations modulo primes up to 31 for the first time. In particular, he has computed the representations attached to a newform with non-rational (but of course algebraic) coefficients, which had never been done before. These computations take place in the Jacobians of modular curves of genus up to 26.