Jean-Marc Couveignes

Action of Modular Correspondences around CM Points

We study the action of modular correspondences in the p- adic neighborhood of CM points. We deduce and prove two stable and efficient p-adic analytic methods for computing singular values of modular functions. On the way we prove a non trivial lower bound for the density of smooth numbers in imaginary quadratic rings and show that the canonical lift of an elliptic curve over Fq can be computed in probabilistic time ? exp((log q)1/2+?) under GRH. We also extend the notion of canonical lift to supersingular elliptic curves and show how to compute it in that case.

Elliptic periods for finite fields

We construct two new families of basis for finite field extensions. Bases in the first family, the so-called elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the so-called normal elliptic bases are normal bases and allow fast (quasi-linear) arithmetic. We prove that all extensions admit models of this kind.

Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujans 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 (Schoofs 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, Ramanujans 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 program. 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 program. The computation of the Galois representations uses their realization, 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 uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Linearizing torsion classes in the Picard group of algebraic curves over finite fields

We address the problem of computing in the group of

Galois invariant smoothness basis

\ell^k

The geometry of flex tangents to a cubic curve and its parameterizations

-torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations.

Computing functions on Jacobians and their quotients

This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field

Boundary of Hurwitz Spaces and Explicit Patching

\bK

Quelques revêtements définis sur ℚ

, one looks for a smoothness basis for

Continuous Approximation of a Discrete Situated and Reactive Multi-agent System: Contribution to Agent Parameterization

\bK^*