Moduli stacks of crystals and isocrystals
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909586198364160 |
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| author | Oh, Gyujin Shimizu, Koji |
| author_facet | Oh, Gyujin Shimizu, Koji |
| contents | Given a liftable smooth proper variety over $\mathbb{F}_p$, we construct the moduli stacks of crystals and isocrystals on it. We show that the former is a formal algebraic stack over $\mathbb{Z}_p$ and the latter is an adic stack -- Artin stack in rigid geometry -- over $\mathbb{Q}_p$. Both stacks come equipped with the Verschiebung endomorphism $V$ corresponding to the Frobenius pullback of (iso)crystals. We study the geometry of the $V$-fixed points over the open substack of irreducible isocrystals, which we use to geometrically count the rank one $F$-isocrystals. Along the way, we carefully develop the theory of adic stacks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moduli stacks of crystals and isocrystals Oh, Gyujin Shimizu, Koji Number Theory Algebraic Geometry Given a liftable smooth proper variety over $\mathbb{F}_p$, we construct the moduli stacks of crystals and isocrystals on it. We show that the former is a formal algebraic stack over $\mathbb{Z}_p$ and the latter is an adic stack -- Artin stack in rigid geometry -- over $\mathbb{Q}_p$. Both stacks come equipped with the Verschiebung endomorphism $V$ corresponding to the Frobenius pullback of (iso)crystals. We study the geometry of the $V$-fixed points over the open substack of irreducible isocrystals, which we use to geometrically count the rank one $F$-isocrystals. Along the way, we carefully develop the theory of adic stacks. |
| title | Moduli stacks of crystals and isocrystals |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2504.14801 |