Arithmetic intersections on non-split Cartan modular curves
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arXiv
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| Format: | Preprint |
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2026
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| author | Love, Jonathan Studnia, Elie Vonk, Jan |
| author_facet | Love, Jonathan Studnia, Elie Vonk, Jan |
| contents | Let $p$ be a prime number, and let $Δ_1,Δ_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{Δ_1})$ and $\mathbb{Q}(\sqrt{Δ_2})$ 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06963 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arithmetic intersections on non-split Cartan modular curves Love, Jonathan Studnia, Elie Vonk, Jan Number Theory 11G18 (Primary) 11G05, 14H10, 11R52 (Secondary) Let $p$ be a prime number, and let $Δ_1,Δ_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{Δ_1})$ and $\mathbb{Q}(\sqrt{Δ_2})$ 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]. |
| title | Arithmetic intersections on non-split Cartan modular curves |
| topic | Number Theory 11G18 (Primary) 11G05, 14H10, 11R52 (Secondary) |
| url | https://arxiv.org/abs/2604.06963 |