Cuspidal divisor class groups of non-split Cartan modular curves

TL;DR

This paper provides an explicit description of modular units and the cuspidal divisor class group for certain non-split Cartan modular curves, including formulas involving Bernoulli numbers to compute their size.

Contribution

It offers a new explicit description of modular units and the structure of the cuspidal divisor class group for non-split Cartan modular curves, extending previous theoretical understanding.

Findings

Explicit formulas for modular units in terms of Siegel functions.

Description of the cuspidal divisor class group as a module over a group ring.

A formula involving Bernoulli numbers for the size of the class group.

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 $|\mathfrak{C}^+_{ns}(p^k)|$.