Computing functions on Jacobians and their quotients

We show how to efficiently compute functions on jacobian varieties and their quotients. We deduce a quasi-optimal algorithm to compute $(l,l)$ isogenies between jacobians of genus two curves

Galois invariant smoothness basis

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 \bK, one looks for a smoothness basis for \bK^* that is left invariant by automorphisms of \bK. For a broad class of finite fields, we manage to construct models that allow such a smoothness basis. This work aims at accelerating discrete logarithm computations in such fields. We treat the cases of codimension one (the linear sieve) and codimension two (the function field sieve)

The geometry of some parameterizations and encodings

We explore parameterizations by radicals of low genera algebraic curves. We prove that for $q$ a prime power that is large enough and prime to $6$, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterized by $3$-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to $2$ modulo $3$. We extend this construction to parameterizations by $\ell$-radicals for small odd integers $\ell$, and make it explicit for $\ell=5$

Approximate computations with modular curves

This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory

Explicit Riemann-Roch spaces in the Hilbert class field

Let $\mathbf K$ be a finite field, $X$ and $Y$ two curves over $\mathbf K$, and $Y\rightarrow X$ an unramified abelian cover with Galois group $G$. Let $D$ be a divisor on $X$ and $E$ its pullback on $Y$. Under mild conditions the linear space associated with $E$ is a free ${\mathbf K}[G]$-module. We study the algorithmic aspects and applications of these modules