Prismatic crystals and $p$-adic Riemann--Hilbert correspondence

Hui Gao; Yu Min; Yupeng Wang

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arXiv · 2411.18780

Prismatic crystals and $p$-adic Riemann--Hilbert correspondence

Hui Gao·Yu Min·Yupeng Wang

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Abstract

We systematically study relative and absolute ${\Delta}_{\mathrm{dR}}^+$-crystals on the (log-) prismatic site of a smooth (resp.~ semi-stable) formal scheme. Using explicit computation of stratifications, we classify (local) relative crystals by certain nilpotent connections, and classify (local) absolute crystals by certain enhanced connections. By using a $p$-adic Riemann--Hilbert functor and an infinite dimensional Sen theory over the Kummer tower, we globalize the results on absolute crystals and further classify them by certain small (global) $\mathbb{B}_{\mathrm{dR}}^+$-local systems.

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Hui Gao, Yu Min, Yupeng Wang. 2024-11-27. Prismatic crystals and $p$-adic Riemann--Hilbert correspondence. https://arxiv.org/abs/2411.18780

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