Section: New Results

Complex multiplication and modularity

Participants : Jean-Marc Couveignes, Andreas Enge, Damien Robert.

The book [16] edited by J.-M. Couveignes and B. Edixhoven, with contributions by J.-M. Couveignes, B. Edixhoven, R. de Jong, F. Merkl and J. Bosman, describes the first polynomial time algorithms for computing Galois representations and coefficients of modular forms. Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's $\tau$-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's $\tau$ of a prime number $p$ can be computed in time bounded by a fixed power of the logarithm of $p$. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands programme. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands programme.

The computation of the Galois representations uses their realisation, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision – in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed – are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformisation and another one using geometry over finite fields.

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 [24] . 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.

With J.-C. Faugère and D. Lubicz, D. Robert has given an explicit construction for a modular correspondance between abelian varieties [14] . This correspondance describes the algebraic relations of ThetaNullWerte of different levels on isogenous abelian varieties. With R. Cosset, D. Robert has then given an algorithm explaining how to construct the corresponding isogeny, when we are given its (maximally isotropic) kernel [20] . This usse a formula by Koizumi for changing the level of the ThetaNullWerte. This is the first algorithm allowing to compute in polynomial time an isogeny between abelian varieties, and a public implementation is available in AVIsogenies .

With K. Lauter, D. Robert has worked on improving the computation of class polynomials in genus 2 by the CRT method. This involves some improvements to detect if the curve is maximal, a better sieving of the primes used, and the use of the CRT over the real quadratic field rather than over $\mathbb{Q}$ for the case of dihedral CM fields. The main improvements comes from using the above isogeny computation, both in order to be able to find a maximal curve from a curve in the correct isogeny class, and in order to find all others maximal curves from one. A preprint describing these improvements is being written, some details are described in the talk http://www.normalesup.org/~robert/pro/publications/slides/2011-04-C2.pdf .

With Reynald Lercier, J.-M. Couveignes has given in [23] 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.