Section: New Results

Complex multiplication and modularity

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

H. Ivey-Law has been implementing efficient algorithms to compute Hilbert class polynomials and modular polynomials for various modular functions, as well as various supplementary algorithms required by, or based on, these two primary components. These algorithms form an important and time-critical part of algorithms used to select elliptic curves for use in cryptographic applications.

The implementation is based on algorithms for these tasks published by A. Sutherland and his collaborators. It includes, more specifically, algorithms to compute Hilbert class polynomials for various different modular functions over $\mathbb{Z}$ or $\mathbb{Z}/M\mathbb{Z}$, modular polynomials for various different modular functions over $\mathbb{Z}$, $\mathbb{Z}/M\mathbb{Z}$, and/or pre-instantiated at a particular point. The supplementary algorithms include functionality for computing equations for isogenies between elliptic curves and equations for their codomains, for manipulating, interrogating and traversing isogeny volcanoes, for computing minimal polycyclic presentations of abstract groups, for testing supersingularity of $j$-invariants, for accessing optimised equations of the modular curve $X_1\left(N\right)$ for $N\le 50$, for finding elliptic curves with a given trace or a given endomorphism ring, for calculating the endomorphism ring of a given elliptic curve, for computing the action of the torsor $Cl\left(𝒪\right)$ on the set of elliptic curves with endomorphism ring $𝒪$ and for enumerating the kernel of the map $Cl(\mathbb{Z}+N𝒪)\to Cl\left(𝒪\right)$.

These algorithms are implemented in an experimental branch of Pari/Gp , and will be integrated in the public version soon.

A. Enge and R. Schertz determine in [13] under which conditions singular values of multiple $\eta$-quotients of square-free level, not necessarily prime to 6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. It turns out that the singular values lie in subfields of the ring class fields of index $2^{k^{\text{'}}-1}$ when $k^{\text{'}}\ge 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on $X_0^{+}\left(p\right)$ for $p$ prime and ramified.

The paper of R. Cosset and D. Robert [25] presenting an algorithm for computing isogenies between principally polarised abelian surface has been accepted for publication in Mathematics of Computation. This paper explains, given the theta coordinates of the points of a maximal isotropic kernel of the $\ell$-torsion, how to compute the corresponding isogeny. It also gives formulæ for the conversion between theta coordinates and Mumford coordinates.

The paper by K. Lauter and D. Robert on Improved CRT Algorithm for Class Polynomials in Genus 2, which was presented at the ANTS-X conference, was published in [18] .

A. Enge and E. Thomé describe in [14] 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 $\vartheta$-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.

N. Mascot's article on computing modular Galois representations [15] has been published in Rendiconti del Circolo Matematico di Palermo. This article describes an algorithm to compute Galois representations attached to a newform, and to deduce the Fourier coefficients of this newform modulo a small prime.

E. Milio has implemented R. Dupont's algorithms  [38] in Pari/Gp . With them, he has calculated the three modular polynomials in genus 2 and level 2 defined by Streng's version of Igusa modular forms and a modular polynomial of genus 2 and level 3 coming from theta modular forms.