  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 very 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
Derickx and 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 Yang.
