Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and $\mathbb{A}^2 \subset \mathbb{P}^2$ the affine and projective planes over $\mathbb{F}_q$, and let $C \subset \mathbb{P}^2$ be a plane projective absolutely irreducible reduced curve. Let $\mathcal{X}$ be its smooth projective model and $\mathcal{J}$ the Jacobian variety of $\mathcal{X}$.
Let $g$ be the genus of $\mathcal{X}$ and $d$ the degree of $C$.

We assume we are given the numerator of the zeta function of the function field $\mathbb{F}_q(\mathcal{X})$. So we know the characteristic polynomial of the Frobenius endomorphism $F_q$ of $\mathcal{J}$. This is a unitary degree $2g$ polynomial $\chi(X)$ with integer coefficients.

Let $\ell \neq p$ 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 $\mathcal{J}\ell^k
$ of $\ell^k$-torsion points in $\mathcal{J}(\mathbb{F}q)$. By {\it nice} we mean that the generating set $(g_i){1 \le i \le I}$ should induce a decomposition of $\mathcal{J}\ell^k
$ as a direct product $\prod_{1 \le i \le I} \langle g_i \rangle$ of cyclic subgroups with non-decreasing orders.

Given such a generating set and an $\mathbb{F}_q$-endomorphism of $\mathcal{J}$, we also want to describe the action of this endomorphism on $\mathcal{J}\ell^k
$ 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.