Generators of the group of modular units for Gamma1(N) over QQ

Abstract

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 for Y(N) can be much simplified and strengthened when applied to Y1(N). The link between the two sets of generators also provides a set of generators of the ring of regular functions of Y1(N), giving a more uniform version of a result of Ja Kyung Koo and Dong Sung Yoon.Comment: 18 page