Arithmetic intersections on non-split Cartan modular curves

Abstract

Let $p$ be a prime number, and let $\hat{I}{”}_1,\hat{I}{”}_2<0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{\hat{I}{”}_1})$ and $\mathbb{Q}(\sqrt{\hat{I}{”}_2})$ 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].

28 pages, 1 table. Comments welcome!