Overview
Input: a modular form f ∈ S2(Γ1(N)) such that Λf is a lattice in C.
We always assume f =
P
σ cσFσ
where F is a newform in S2(Γ1(N)),
{Fσ
} are the Galois conjugates of f , and cσ ∈ Q.
Goals:
• Compute the Q-curve Ef in Weierstrass form.
• Determine if Ef can be parametrized by modular units.
• If so, compute ϕ in algebraic form. By this we mean finding two
modular units u, v ∈ Q(X1(N)) such that Q(Ef ) ∼
= Q(u, v).
We will construct u and v using Siegel units
ga,b(τ) = qα
Y
n≥0
n≡a mod N
(1 − qn/N
ζb
N )
Y
n≥1
n≡−a mod N
(1 − qn/N
ζ−b
N ).
where a, b ∈ Z/NZ, α = B2({a/N}), qα
= e2πiατ
, ζN = e2πi/N
.
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