Modular curves
Curves Points
Pairs (E, α)
X(N)

where α : (Z/NZ)2
' E[N]
and eN(α(1, 0), α(0, 1)) = ζN
Pairs (E, P)
X1(N)

where P ∈ E
has exact order N
X(1) Elliptic curves E
where ζN is a fixed primitive N-th root of 1.
Nicolas Mascot p-adic computation of Galois representations