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arXiv · 2608.19612

Arithmetic partial differential operators on ramified extensions of $\ZZ_p$

Marshall Donn·Lance Edward Miller·William D. Taylor
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Abstract

The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over $\ZZ_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over $\ZZ_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_\pi$ with uniformizer $\pi$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $\pi$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.

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Marshall Donn, Lance Edward Miller, William D. Taylor. 2026-08-20. Arithmetic partial differential operators on ramified extensions of $\ZZ_p$. https://arxiv.org/abs/2608.19612

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