Step 6: Certifying the parametrization
Each hi is of the form
C
Y
a,b∈Z/NZ
g
ea,b
a,b (C ∈ Q(ζN )×
, ea,b ∈ Z).
4. Prove that these products are indeed modular for Γ1(N) (in general,
such a product is modular for Γ(12N2
) only). This uses a criterion
of Kubert–Lang.
5. Denoting by u1, u2 these modular units, prove that P(u1, u2) = 0.
For this, check the q-expansion to high enough accuracy.
The data (P, u1, u2) certifies the modular parametrisation.
We can also certify the images of the cusps computed previously.
Question. How to describe and certify a modular parametrization when
no modular unit is available?
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