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

TL;DR

This paper provides two explicit generator sets for the group of invertible regular functions on the modular curve Y1(N), one based on defining equations of smaller modular curves and the other on Siegel functions, confirming a conjecture and generalizing previous results.

Contribution

It introduces a surprising set of generators related to defining equations of Y1(k) for k ≤ N/2, proving a conjecture, and also offers a generator set using classical Siegel functions, expanding the understanding of modular units.

Findings

Confirmed a conjecture of Derickx and Van Hoeij.

Established a set of generators related to defining equations of Y1(k).

Connected generators to elliptic divisibility sequences and Siegel functions.

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 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.