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} $.