An archimedean approach to singular moduli on Shimura curves

TL;DR

This paper provides a new proof for a generalization of Gross and Zagier's work on singular moduli, using Green's functions on Shimura curves instead of $p$-adic methods, inspired by analytic techniques.

Contribution

It introduces an alternative, Green's function-based proof for the Shimura curve generalization, highlighting differences from the $p$-adic approach.

Findings

Green's function evaluation at CM points on Shimura curves

Comparison of analytic and $p$-adic proof techniques

Validation of the generalized singular moduli formula

Abstract

We give a new proof of a recent generalization to Shimura curve of genus 0 of the work of Gross and Zagier in `On singular moduli'. This generalization was conjectured by Giampietro and Darmon and proved by Daas by using $p$-adic $\Theta$-functions as an analogue of the $j$-invariant. Instead of working $p$-adically, we prove this result by evaluating Green's function at CM points on the Shimura curve. Our strategy is inspired by the analytic proof of Gross--Zagier. We put a special emphasis on both the similarities and the differences with the $p$-adic proof.