An archimedean approach to singular moduli on Shimura curves

Abstract

Abstract We give a new proof of a recent generalization to Shimura curves of genus 0 of the work of Gross and Zagier in their paper ‘On singular moduli’. This generalization was conjectured by Giampietro and Darmon and proved by Daas by using p -adic $$\Theta $$ Θ -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 and Zagier. We put a special emphasis on both the similarities and the differences with the p -adic proof.