Cuspidal divisor class groups of non-split Cartan modular curves (1605.00375v1)

Abstract: I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ 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 $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/{\pm 1}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,\chi}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.