Q-curves and modular forms
Let ϕ: X1(N)Q → E be a modular parametrization.
Then ϕ∗
(ωE ) = ωf = 2πif (τ)dτ for some f ∈ S2(Γ1(N)) (not necessarily
a newform!). Moreover
Λf :=
nZ
γ
ωf : γ ∈ H1(X1(N), Z)
o
is a lattice in C, and we have E(C) ∼
= C/Λf .
Conversely, let f ∈ S2(Γ1(N)) such that Λf is a lattice in C. Then
Ef = C/Λf is a Q-curve with modular parametrization
ϕ: X1(N)Q → Ef , τ 7→
hZ τ
0
ωf
i
.
Questions. Given E, can we compute f , and conversely? Can we
compute ϕ? (And what does this mean?)
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