Arithmetic of quaternion origami

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...

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Bibliographic Details
Published in:arXiv.org 2018-05
Main Authors: Davis, Rachel, Edray Herber Goins
Format: Article
Language:English
Subjects:
Curves
Division
Isomorphism
Mathematical analysis
Polynomials
Quaternions
Online Access:Get full text
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Summary: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.
ISSN:2331-8422