Arithmetic intersections on non-split Cartan modular curves

TL;DR

This paper computes arithmetic intersection numbers of CM divisors on non-split Cartan modular curves at all finite primes, extending known results from split cases and providing a new moduli interpretation at bad reduction primes.

Contribution

It introduces a moduli interpretation for the smooth locus of the regular model of non-split Cartan modular curves, enabling intersection calculations at primes of bad reduction.

Findings

Determined intersection numbers at primes of bad reduction for non-split Cartan curves.

Extended Gross--Kohnen--Zagier results to inert prime cases.

Provided a new moduli interpretation for the regular model at bad primes.

Abstract

Let $p$ be a prime number, and let $\Delta_1,\Delta_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{\Delta_1})$ and $\mathbb{Q}(\sqrt{\Delta_2})$ the height pairings of the corresponding CM divisors on $X_{\mathrm{spl}}^+(p)$ were determined by Gross--Kohnen--Zagier [GKZ87]. When $p$ is inert, we determine the arithmetic intersection numbers of the corresponding divisors on $X_{\mathrm{ns}}^+(p)$ at all finite primes. The key point of our analysis is at the prime of bad reduction $p$: to determine the intersection numbers at $p$, we provide a moduli interpretation for the smooth locus in the regular model of $X_{\mathrm{ns}}^+(p)$ over $\mathrm{Spec}(\mathbb{Z})$ constructed by Edixhoven--Parent [EP24].