Makdisi’s algorithms
All we need is the matrix
V =



v1(P1) v2(P1) · · ·
.
.
.
.
.
.
v1(Pn) v2(Pn) · · ·



where v1, v2 are “functions” on C forming a basis of the space
of global sections of a line bundle L on C (≈ Riemann-Roch
space), and P1, P2, · · · ∈ C are sufficiently many points.
A point on J is then represented by a matrix
W =



w1(P1) w2(P1) · · ·
.
.
.
.
.
.
w1(Pn) w2(Pn) · · ·



where w1, w2, · · · is a basis of a subspace.
Nicolas Mascot p-adic computation of Galois representations