Section: New Results

Complex multiplication

Participant : Andreas Enge.

A. Enge's article analysing and comparing the complexity of algorithms computing complex multiplication elliptic curves and ring class fields of imaginary-quadratic orders [17] has appeared in print. The new algorithm of quasi-linear complexity (that is, linear up to logarithmic factors) in the size of the output class polynomial has been implemented in the Cm software, see 5.5 , relying on the helper libraries Mpfrcx , see 5.4 , and Mpc , see 5.3 ; parts of the algorithm have also been included into the development version of Pari/Gp , see 5.1 . The results are summarised in an overview article aimed at the computer algebra community [18] .

With F. Morain, A. Enge has determined exhaustively under which conditions “generalised Weber functions”, that is, simple quotients of functions of not necessarily prime transformation level and not necessarily of genus 1, yield class invariants [28] . The result is a new infinite family of generators for ring class fields, usable to determine complex multiplication curves. We 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.

In the same vein as the result for univariate class polynomials, [16] proposes a quasi-linear algorithm to compute bivariate modular polynomials, which are at the heart of modern point counting algorithms for elliptic curves. The algorithm relies on asymptotically fast evaluation and interpolation. Its unpublished implementation has been used to compute polynomials of degree around 10000, each filling 16 GB of disk space. This has enabled the current point counting record for a curve of 2500 decimal digits [35] .