Section: New Results

Complex multiplication and modularity

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

The article by D. Lubicz and D. Robert which explains how to compute an isogeny between two abelian varieties given the kernel (but with different levels of theta structures) has been published [16] . The preprint [25] with R. Cosset and D. Robert extends these method to provide an algorithm constructing the corresponding isogeny without changing the level. This give the first algorithm allowing to compute in polynomial time an isogeny between abelian varieties, and a public implementation is available in AVIsogenies . The drawback of this algorithm is that it needs the geometric points of the kernel. To compute an isogeny of degree ${\ell }^g$ over a finite field, working with geometric points requires to take an extension of degree up to ${\ell }^g-1$, and the situation is much worse over a number field. Recently, D. Lubicz and D. Robert have explained how to compute the corresponding isogeny given only the equations of the kernel. This gives a quasi-linear algorithm (in the degree ${\ell }^g$ of the isogeny) when $\ell$ is congruent to 1 modulo 4.

With K. Lauter, D. Robert has worked on improving the computation of class polynomials in genus 2 by the CRT method. The main improvements come from using the above isogeny computation, both to find a maximal curve from a curve in the correct isogeny class, and to find all other maximal curves from one. Further improvements are in the endomorphism ring computation to detect if the curve is maximal, a better sieving of the primes used (and a dynamic selection of them), and the use of the CRT over the real quadratic field rather than over $\mathbb{Q}$ for the case of dihedral CM fields to find factors of the class polynomials. These results have been published at the ANTS conference [30] .

With C. Ritzenthaler, Damien Robert has shown how to compute explicitly the Serre obstruction for abelian varieties isogenous to a product of three elliptic curves. This allows to find genus 3 curves with many points over a finite field. The corresponding code has been implemented in an (experimental) version of AVIsogenies .

In [24] , H. Cohen studies several methods for the numerical computation of Petersson scalar products. In particular he proves a generalisation of Haberland's formula to any subgroup of finite index $G$ of $\Gamma ={PSl}_2\left(Z\right)$, which gives a fast method to compute these scalar products when a Hecke eigenbasis is not necessarily available.

J.-M. Couveignes and B. Edixhoven explore in [19] the relevance of numerical methods in dealing with higher genus curves and their Jacobians. Fast exponentiation is crucial in this context as a stable substitute to Newton's method and analytic continuation. Arakelov theory provides the necessary complexity estimates.

With Reynald Lercier, J.-M. Couveignes has given in [26] a quasi-linear time randomised algorithm that on input a finite field ${𝔽}_q$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in ${𝔽}_q\left[x\right]$. The running time is $d^{1+o\left(1\right)}\times {(logq)}^{5+o\left(1\right)}$ elementary operations. The $o\left(1\right)$ in $d^{1+o\left(1\right)}$ is a function of $d$ that tends to zero when $d$ tends to infinity. And the $o\left(1\right)$ in ${(logq)}^{5+o\left(1\right)}$ is a function of $q$ that tends to zero when $q$ tends to infinity. The fastest previously known algorithm for this purpose was quadratic in the degree. The algorithm relies on the geometry of elliptic curves over finite fields (complex multiplication) and on a recent algorithm by Kedlaya and Umans for fast composition of polynomials.

In [32] , N. Mascot shows how to compute modular Galois representations associated with a newform $f$ and the coefficients of $f$ modulo a small prime $\ell$. To this end, he designs a practical variant of the complex approximation method presented in the book edited by B. Edixhoven and J.-M. Couveignes [8] . Its efficiency stems from several new ingredients. For instance, he uses fast exponentiation in the modular Jacobian instead of analytic continuations, which greatly reduces the need to compute abelian integrals, since most of the computation handles divisors. Also, he introduces an efficient way to compute arithmetically well-behaved functions on Jacobians. He illustrates the method on the newform $\Delta$, and manages to compute for the first time the associated faithful representation modulo $\ell$ and the values modulo $\ell$ of Ramanujan's $\tau$ function at huge primes for $\ell \in \{11,13,17,19\}$. In particular, he gets rid of the sign ambiguity stemming from the use of a non-faithful representation as in J. Bosman's work.

A. Enge and R. Schertz determine in [29] 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.

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 [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.