Makdisi’s moduli-friendly forms
α : (Z/NZ)2
' E[N]
For v, w ∈ (Z/NZ)2
such that v, w, v + w are all nonzero, let
λv,w : (E, α) 7−→ slope of line joining α(v) to α(w).
Theorem (Makdisi, 2011)
1 λv,w is a modular form of weight 1 for X(N).
Nicolas Mascot p-adic computation of Galois representations