Sub-representations of ρJ,`
Frequently, we only want the representation ρT coming from
the points of a Galois-stable F`-subspace T ⊂ J[`].
Given p ∈ N prime, let
L(x) = det(x − Frobp |J[`]) and χT (x) = det(x − Frobp |T ),
so that χT | L.
If χT is coprime with ψT = L/χT , then we can generate
random points of T by applying ψT (Frobp) to random points
of J[`]
We can compute ρT .
Nicolas Mascot p-adic computation of Galois representations