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).
2 The R-algebra generated by the λv,w contains all modular
forms for X(N), except cuspforms of weight 1.
3 The λv,w are moduli-friendly!
We can compute in the Jacobian of X(N) without
equations nor q-expansions, just by looking at E[N] for one E!
Nicolas Mascot p-adic computation of Galois representations