Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

We find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{\rm ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The cuspidal divisor class group $\mathfrak{C}^+_{\rm ns}(p^k)$ on $X^+_{\rm ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. We give a formula for $|\mathfrak{C}^+_{\rm ns}(p^k)|$ involving generalized Bernoulli numbers $B_{2,\chi}$.