Liftings of Reduction Maps for Quaternion Algebras

TL;DR

This paper develops a method to lift reduction maps from CM points to supersingular points in quaternion algebras, establishing a detailed correspondence between CM points on indefinite and definite quaternion algebras.

Contribution

It introduces a new lifting technique for reduction maps in quaternion algebras, linking CM points across different algebra types with explicit correspondence.

Findings

Constructed liftings of reduction maps for quaternion algebras.

Established a correspondence between CM points on indefinite and definite quaternion algebras.

Applied the method to relate CM points with given conductors across algebra types.

Abstract

We construct liftings of reduction maps from CM points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras.