Finiteness of function field-valued points on exceptional Shimura varieties
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915634511609856 |
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| author | Bakker, Benjamin Shankar, Ananth N. Tsimerman, Jacob |
| author_facet | Bakker, Benjamin Shankar, Ananth N. Tsimerman, Jacob |
| contents | Let $C/k$ be a smooth curve over a finite field of characteristic $p>0$. We prove that there are finitely many principally polarized abelian schemes of given dimension $g$ over $C$ up to $p$-power isogeny. For curves over $\overline{k}$, we prove that the moduli space of such abelian schemes is finite type up to $p$-power isogeny.
Moreover, we generalize this result to arbitrary (not necessarily abelian type) Shimura varieties $S$ and sufficiently large primes $p$ in terms of $S$: The space of generically ordinary morphisms $C\to S_{k}$ (resp. $C\to S_{\overline{k}})$ is finite (resp. finite type) up to $p$-Hecke orbits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finiteness of function field-valued points on exceptional Shimura varieties Bakker, Benjamin Shankar, Ananth N. Tsimerman, Jacob Number Theory Algebraic Geometry Let $C/k$ be a smooth curve over a finite field of characteristic $p>0$. We prove that there are finitely many principally polarized abelian schemes of given dimension $g$ over $C$ up to $p$-power isogeny. For curves over $\overline{k}$, we prove that the moduli space of such abelian schemes is finite type up to $p$-power isogeny. Moreover, we generalize this result to arbitrary (not necessarily abelian type) Shimura varieties $S$ and sufficiently large primes $p$ in terms of $S$: The space of generically ordinary morphisms $C\to S_{k}$ (resp. $C\to S_{\overline{k}})$ is finite (resp. finite type) up to $p$-Hecke orbits. |
| title | Finiteness of function field-valued points on exceptional Shimura varieties |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2511.18206 |