We give two explicit sets of generators of the group of invertible regular functions over on the modular curve .
The first set of generators is very surprising. It is essentially the set of defining equations of for when all these modular curves are simultaneously embedded into the affine plane, and this proves a conjecture of Derickx and Van Hoeij [DvH14]. This set of generators is an elliptic divisibility sequence in the sense that it satisfies the same recurrence relation as the elliptic division polynomials.
The second set of generators is explicit in terms of classical analytic functions known as Siegel functions. This is both a generalization and a converse of a result of Yang [Yan04, Yan09].
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