## Generators of the group of modular units for Gamma1(N) over QQ. (arXiv:1503.08127v2 [math.NT] UPDATED)

We give two explicit sets of generators of the group of invertible regular
functions over QQ on the modular curve Y1(N).
The first set of generators is the most surprising. It is essentially the set
of defining equations of Y1(k) for k<=N/2 when all these modular curves are
simultaneously embedded into the affine plane, and this proves a conjecture of
Maarten Derickx and Mark van Hoeij. 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 Yifan Yang.
Our proof consists of two parts. First, we relate our two sets of generators.
Second, we use q-expansions and Gauss' lemma for power series to prove that our
functions generate the full group of modular functions. This second part shows
how a proof of Kubert and Lang查看全文