Heights of Special Points on Quaternionic Shimura Varieties

TL;DR

This paper derives an explicit formula for the heights of special points on quaternionic Shimura varieties, relating them to Faltings heights of CM abelian varieties, and demonstrates their compatibility with canonical heights, providing bounds related to CM field discriminants.

Contribution

It provides a new explicit height formula for special points on quaternionic Shimura varieties and shows compatibility with existing canonical height definitions.

Findings

Derived an explicit height formula in terms of Faltings heights.

Proved compatibility with the canonical height of partial CM-types.

Established subpolynomial bounds on heights relative to CM field discriminants.

Abstract

Let $B/F$ be a quaternion algebra over a totally real number field. We give an explicit formula for heights of special points on the quaternionic Shimura variety associated with $B$ in terms of Faltings heights of CM abelian varieties. Special points correspond to CM fields $E$ and partial CM-types $\phi \subset \mathrm{Hom}(E, \mathbb{C})$. We then show that our height is compatible with the canonical height of a partial CM-type defined by Pila, Shankar, and Tsimerman. This gives another proof that the height of a partial CM-type is bounded subpolynomially in terms of the discriminant of $E$.