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arXiv · 2603.08569

An archimedean approach to singular moduli on Shimura curves

Mateo Crabit Nicolau
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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.

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Mateo Crabit Nicolau. 2026-03-09. An archimedean approach to singular moduli on Shimura curves. https://arxiv.org/abs/2603.08569

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