Step 6: Certifying the parametrization
Because our computations were numerical, we haven’t proved the
parametrization exists yet!
1. Compute the q-expansion of ϕ∗
(x) and ϕ∗
(y) in Q(ζN )((q)).
(
x = u2
q−2e
+ O(q−2e+1
)
y = u3
q−3e
+ O(q−3e+1
)
with e = eϕ(∞) and u ∈ Q(ζN )×
, exactly as in elltaniyama:
using the two equations y2
= x3
− 27c4x − 54c6 and ωf = dx/2y,
determine inductively the Fourier coefficients of x and y.
2. Deduce the q-expansion of h1 and h2.
3. Express h1, h2 as products of Siegel units by comparing the divisors
and checking the leading coefficient.
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