Arithmetic properties of orders in imaginary quadratic fields

TL;DR

This paper explores the arithmetic properties of orders in imaginary quadratic fields, focusing on ray class fields, Galois representations of CM elliptic curves, form class groups, and L-functions, revealing new insights into their interconnected structures.

Contribution

It introduces new results on the structure and properties of ray class fields and related Galois representations for orders in imaginary quadratic fields.

Findings

Characterization of ray class fields for orders

Connections between Galois representations and complex multiplication

Results on form class groups and L-functions for orders

Abstract

Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to $K_{\mathcal{O},\,N}$, mainly about Galois representations attached to elliptic curves with complex multiplication, form class groups and $L$-functions for orders.