Birational geometry of quaternions

TL;DR

This paper investigates the algebraic surfaces associated with quaternion algebras and their Hilbert class fields, revealing birational relationships and applying these findings to the function field analogy.

Contribution

It establishes that the avatar of the Hilbert class field of a quaternion algebra is birationally equivalent to the avatar of the algebra itself, advancing understanding of their geometric structures.

Findings

The avatar of the Hilbert class field is obtained from the avatar of the quaternion algebra via a birational map.

The study provides new insights into the geometric structure of quaternion algebra avatars.

Application of results to the function field analogy enhances theoretical understanding.

Abstract

The Hilbert class field of the quaternion algebra $B$ is an algebra $\mathscr{H}(B)$ such that every two-sided ideal of $B$ is principal in $\mathscr{H}(B)$. We study the avatars of $B$ and $\mathscr{H}(B)$, i.e. algebraic surfaces attached to the quaternion algebras. It is proved that the avatar of $\mathscr{H}(B)$ is obtained from the avatar of $B$ by a birational map. We apply this result to the function field analogy.