Arithmetic intersections on non-split Cartan modular curves

Let $p$ be a prime number, and let ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2<0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{{\mathrm{\Delta }}_1})$ and $\mathbb{Q}(\sqrt{{\mathrm{\Delta }}_2})$ the height pairings of the corresponding CM divisors on $X_{\mathrm{s}\mathrm{p}\mathrm{l}}^{+}(p)$ were determined by Gross–Kohnen–Zagier [GKZ87]. When $p$ is inert, we determine the arithmetic intersection numbers of the corresponding divisors on $X_{\mathrm{n}\mathrm{s}}^{+}(p)$ 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 $X_{\mathrm{n}\mathrm{s}}^{+}(p)$ over $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathbb{Z})$ constructed by Edixhoven–Parent [EP24].