Modular Heights of Quaternionic Shimura Curves

TL;DR

This paper derives a formula linking the modular height of quaternionic Shimura curves over totally real fields to the logarithmic derivative of the Dedekind zeta function, building on advanced number theory and height formulas.

Contribution

It provides a new explicit formula for the modular height of quaternionic Shimura curves, connecting geometric invariants with special values of zeta functions.

Findings

Established a formula relating modular height to Dedekind zeta derivative

Extended Gross-Zagier and Colmez conjecture techniques to quaternionic Shimura curves

Connected arithmetic geometry with analytic number theory through this formula

Abstract

The goal of this paper is to prove a formula expressing the modular height of a quaternionic Shimura curve over a totally real number field in terms of the logarithmic derivative of the Dedekind zeta function of the totally real number field. Our proof is based on the work of Yuan-Zhang-Zhang on the Gross-Zagier formula and the work of Yuan-Zhang on the averaged Colmez conjecture. All these works are in turn inspired by the Pioneering work of Gross-Zagier and some philosophies of Kudla's program.