Section: New Results

Complex multiplication

Participants : Andreas Enge, François Morain.

A. Enge has been able to analyse precisely the complexity of class polynomial computations via complex floating point approximations [4] . Using techniques from fast symbolic computation, namely multievaluation of polynomials, and results from R. Dupont's PhD thesis [32] , he has obtained two algorithms which are quasi-linear (up to logarithmic factors) in the output size. The second algorithm has been used for a record computation of a class polynomial of degree 100,000, the largest coefficient of which has almost 250,000 bits. The implementation is based on GMP , mpfr, mpc and mpfrcx (see Section 5); the only limiting factor for going further has become the memory requirements of the final result.

Alternative algorithms use p -adic approximations or the Chinese remainder theorem to compute class polynomials over the integers. A. Enge and his coauthors have presented an optimized algorithm based on Chinese remaindering in [2] and improved the number theoretic bounds underlying the complexity analysis. They have shown that all three different approaches have a quasi-linear complexity, while the the floating point algorithm appeared to be the fastest one in practice.

Inspired by [2] , A. Sutherland has come up with a new implementation of the Chinese remainder based algorithm that has led to new record computations [62] . Unlike the other algorithms, this approach does not need to hold the complete polynomial in main memory, but essentially only one coefficient at a time, which enables it to go much further. The main bottleneck is currently an extension of the algorithm to class invariants, which is work in progress by A. Enge.

Morai and Enge have contributed to the study of generalized Weber functions, enabing a partial classification for some cases [21] .