p-adic Hodge theory of unipotent completions revisited

Ruth Wild

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arXiv · 2610.17169

p-adic Hodge theory of unipotent completions revisited

Ruth Wild

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Abstract

Let $K/\mathbb{Q}_p$ be a finite extension, and let $X/K$ be an algebraic variety. We present a new proof that the unipotent completion of the étale fundamental group of $X_{\bar{K}}$ is a pro-de Rham representation, and show that it is pro-semi-stable whenever $X$ has semi-stable étale cohomology. We also reprove, using syntomic methods, that it is pro-crystalline at integral base points whenever $X$ admits a smooth proper formal model. All three proofs use the bar construction and are inspired by Wojtkowiak's approach to unipotent fundamental groups.

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Ruth Wild. 2026-10-12. p-adic Hodge theory of unipotent completions revisited. https://arxiv.org/abs/2610.17169

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