Modular Curves of Composite Level

Andreas Enge (1, 2) , Reinhard Schertz (3)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 TANC - Algorithmic number theory for cryptology
3 Institut für Mathematik [Augsburg]

We examine a class of functions on $X_0 (N)$ where $N$ is the product of two arbitrary primes. The functions are built as products of Dedekind's $\eta$-function and play a role in the construction of elliptic curves with complex multiplication. We show how to determine the modular polynomials relating them to the absolute modular invariant $j$ and prove different properties of these polynomials. In particular, we show that they provide models for the modular curves $X_0 (N)$.

Domaines: Théorie des nombres [math.NT]

Andreas Enge, Reinhard Schertz. Modular Curves of Composite Level. Acta Arithmetica, 2005, 118 (2), pp.129-141. ⟨inria-00386309⟩