Abstract

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

Jean-Marc Couveignes
Université de Toulouse II (Le Mirail)

Let $\Fq$ be a finite field of characteristic $p$ and $\AA^2\subset \PP^2$ the affine and projective planes over $\Fq$ and $C\subset \PP^2$ a plane projective absolutely irreducible reduced curve and $\cX$ its smooth projective model and $\cJ$ the jacobian variety of $\cX$ .
Let $g$ be the genus of $\cX$ and $d$ the degree of $C$ .

We assume we are given the numerator of the zeta function of the function field $\Fq(\cX)$ . So we know the characteristic polynomial of the Frobenius endomorphism $F_q$ of $\cJ$ . This is a unitary degree $2g$ polynomial $\chi(X)$ with integer coefficients.

Let $\ell\not = p$ be a prime integer and let $n=\ell^k$ be a power of $\ell$ . We look for a {\it nice generating set} for the group $\cJ[\ell^k](\Fq)$ of $\ell^k$ -torsion points in $\cJ(\Fq)$ . By {\it nice} we mean that the generating set $(g_i)_{1\le i\le I}$ should induce a decomposition of $\cJ[\ell^k](\Fq)$ as a direct product $\prod_{1\le i\le I} $of cyclic subgroups with non-decreasing orders. Given such a generating set and an$ \Fq $-endomorphism of$ \cJ $, we also want to describe the action of this endomorphism on$ \cJ[\ell^k](\Fq) $by an$ I\times I $integer matrix. These general algorithms are then applied to modular curves in order to compute explicitly the modular representation modulo$ \ell $associated with some modular form (e.g. the discriminant modular form (level$ 1 $and weight$ 12$)).

This makes a connexion with the Edixhoven's program for computing coefficients of modular forms.