Wojciech J. Gajda

Abstract:

The modular curve Y1(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 Y1(N). We then show that the same recurrence relation also gives explicit algebraic formulae for a basis of the group of units on Y1(N), and we generalise to Y(N). This result proves and generalises a conjecture of Maarten Derickx and Mark van Hoeij.