Arithmetic of quaternion origami

TL;DR

This paper investigates the algebraic structure of fields generated by preimages of rational points under a specific origami map with Galois group Q_8, linking these fields to elliptic curve division fields.

Contribution

It provides explicit defining polynomials and analyzes the Galois groups of fields generated by preimages, connecting origami coverings to elliptic curve division fields.

Findings

Explicit polynomial $f_{E, Q_8,P}$ for the field extension.

Isomorphism between subfields of the preimage field and elliptic curve 4-division field.

Analysis of the Galois group structure of the generated fields.

Abstract

We study origami $f: C \rightarrow E$ with $G$-Galois cover $Q_8$. For a point $P \in E(\mathbb{Q}) \backslash \left\{ \mathcal{O} \right\}$, we study the field obtained by adjoining to $\mathbb{Q}$ the coordinates of all of the preimages of $P$ under $f$. We find a defining polynomial, $f_{E, Q_8,P}$, for this field and study its Galois group. We give an isomorphism depending on $P$ between a certain subfield of this field and a certain subfield of the 4-division field of the elliptic curve.