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arXiv · 2504.14801

Moduli stacks of crystals and isocrystals

Gyujin Oh·Koji Shimizu
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Abstract

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.

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Gyujin Oh, Koji Shimizu. 2025-04-21. Moduli stacks of crystals and isocrystals. https://arxiv.org/abs/2504.14801

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