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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.00687 |
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| _version_ | 1866912738292269056 |
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| author | Jiang, Ruofan |
| author_facet | Jiang, Ruofan |
| contents | We study ordinary abelian schemes in characteristic $p$ and their moduli spaces from the perspective of char $p$ Mumford--Tate, log Ax--Lindemann, and geometric André--Oort conjectures (abbreviated as $\MTT_p$, $\mathrm{logAL}_p$ and geoAO$_p$). In this paper, we achieve multiple goals: (\textbf{A}) establish the implication $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p \Rightarrow \mathrm{geoAO_p}$, and show that they all follow from the Tate conjecture for abelian varieties. The equivalence $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p$ is exploited from both sides, which enables us to \noindent(\textbf{B}) develop a representation theory approach to $\mathrm{logAL}_p$ and $\mathrm{geoAO_p}$ by first establishing many cases of MT$_p$ via classical techniques, and (\textbf{C}) develop an algebraization approach to $\MTT_p$ that transcends the limitation of classical methods. In particular, we introduce ``crystalline Hodge loci'', a rigid analytic geometric object that encodes the essential information needed for proving $\mathrm{logAL}_p$, while being very approachable via (integral and relative) $p$-adic Hodge theory. This enables us to prove $\mathrm{logAL}_p$ for compact Tate-linear curves with unramified $p$-adic monodromy. As an application, we establish $\MTT_p$ for many abelian fourfolds of $p$-adic Mumford type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00687 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-adic monodromy and mod $p$ unlikely intersections, II Jiang, Ruofan Number Theory We study ordinary abelian schemes in characteristic $p$ and their moduli spaces from the perspective of char $p$ Mumford--Tate, log Ax--Lindemann, and geometric André--Oort conjectures (abbreviated as $\MTT_p$, $\mathrm{logAL}_p$ and geoAO$_p$). In this paper, we achieve multiple goals: (\textbf{A}) establish the implication $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p \Rightarrow \mathrm{geoAO_p}$, and show that they all follow from the Tate conjecture for abelian varieties. The equivalence $\mathrm{MT}_p\Leftrightarrow \mathrm{logAL}_p$ is exploited from both sides, which enables us to \noindent(\textbf{B}) develop a representation theory approach to $\mathrm{logAL}_p$ and $\mathrm{geoAO_p}$ by first establishing many cases of MT$_p$ via classical techniques, and (\textbf{C}) develop an algebraization approach to $\MTT_p$ that transcends the limitation of classical methods. In particular, we introduce ``crystalline Hodge loci'', a rigid analytic geometric object that encodes the essential information needed for proving $\mathrm{logAL}_p$, while being very approachable via (integral and relative) $p$-adic Hodge theory. This enables us to prove $\mathrm{logAL}_p$ for compact Tate-linear curves with unramified $p$-adic monodromy. As an application, we establish $\MTT_p$ for many abelian fourfolds of $p$-adic Mumford type. |
| title | $p$-adic monodromy and mod $p$ unlikely intersections, II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.00687 |