Arithmetic of quaternion origami

Abstract

We study origami $f:C\to E$ 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.