Algebraic modular forms
Definition
An algebraic modular form of weight k for X(N) over R is a
rule f assigning a value to triples (E/R, α, ω), such that
f (E, α, uω) = u−k
f (E, α, ω)
for all u ∈ R×
.
Short Weierstrass
(E) : y2
= x3
+ Ax + B
ω = dx/2y.
Isomorphic to
(E0
) : y2
= x3
+ A0
x + B0
by (x, y) 7→ (u2
x, u3
y), A0
= u4
A, B0
= u6
B, ω0
= u−1
ω.
Nicolas Mascot p-adic computation of Galois representations