Obtaining modular units via a recurrence relation

Abstract: The modular curve $Y^1(N)$ parametrises pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of order $N$ on $E$. One tool for studying this curve is the group of modular units on it, that is, the group of algebraic functions with no poles or zeroes.

We first review how a recurrence relation (related to elliptic divisibility sequences) gives rise to defining equations for the curves $Y^1(N)$. We then show that the same recurrence relation also gives explicit algebraic formulae for a basis of the group of units on $Y^1(N)$.

This proves a conjecture of Maarten Derickx and Mark van Hoeij.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar