Properties of cuspidal divisor class numbers of non-split Cartan modular curves

TL;DR

This paper investigates the properties and growth behavior of the cuspidal divisor class numbers associated with non-split Cartan modular curves, focusing on their $p$-primary parts.

Contribution

It provides new estimates for the order of growth of the $p$-primary part of the cuspidal divisor class group of these modular curves.

Findings

Estimated the growth rate of $|rak{C}^+_{ns}(p)|$

Analyzed the structure of the $p$-primary component

Contributed to understanding the arithmetic of non-split Cartan modular curves

Abstract

Let $ \mathfrak{C}^+_{ns}(p) $ be the Cuspidal Divisor Class Group of the modular curves $X^+_{ns}(p) $ associated to the normalizer of a non-split Cartan subgroup of level $ p$. I study the $ p-$primary part of $ \mathfrak{C}^+_{ns}(p) $ and estimate the order of growth of $ |\mathfrak{C}^+_{ns}(p)| $.