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}
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.
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