arXiv subjects

Guido Bosco

Publications and source records attributed to Guido Bosco.

4 recordsLinked to original sources

arXiv2025

Analytic de Rham stacks of Fargues-Fontaine curves

We define and initiate the study of analytic de Rham stacks of relative Fargues-Fontaine curves. To this end, we develop a theory of analytic de Rham stacks with sufficiently strong descent and approximation properties. Specializing to the de Rham stack of the Fargues-Fontaine curve attached to $\mathbb{C}_p$, we apply the general theory to obtain a new geometric proof of the $p$-adic monodromy theorem, avoiding any reliance on earlier results on $p$-adic differential equations. Building on the foundations established here, we plan in a sequel to investigate the cohomology of de Rham stacks of relative Fargues-Fontaine curves in geometric situations and, in particular, provide a stack-theoretic definition of Hyodo-Kato cohomology.

math.AG↗

arXiv2024

Torsion in $p$-adic étale cohomology: remarks and conjectures

Let $C$ be a complete algebraically closed extension of $\mathbb{Q}_p$, and let $\mathfrak{X}$ be a smooth formal scheme over $\mathcal{O}_C$. By the work of Bhatt--Morrow--Scholze, it is known that when $\mathfrak{X}$ is proper, the length of the torsion in the integral $p$-adic étale cohomology of the generic fiber $\mathfrak{X}_C$ is bounded above by the length of the torsion in the crystalline cohomology of its special fiber. In this note, we focus on the non-proper case and observe that when $\mathfrak{X}$ is affine, the torsion in the integral $p$-adic étale cohomology of $\mathfrak{X}_C$ can even be expressed as a functor of the special fiber, unlike in the proper case. As a consequence, we show that, surprisingly, if $\mathfrak{X}$ is affine, the integral $p$-adic étale cohomology groups of $\mathfrak{X}_C$ have finite torsion subgroups. We discuss further applications and propose conjectures predicting the torsion in the integral $p$-adic étale cohomology of a broader class of rigid-analytic varieties over $C$.

math.AG↗

arXiv2023

On the $p$-adic pro-étale cohomology of Drinfeld symmetric spaces

Via the relative fundamental exact sequence of $p$-adic Hodge theory, we determine the geometric $p$-adic pro-étale cohomology of the Drinfeld symmetric spaces defined over a $p$-adic field, thus giving an alternative proof of a theorem of Colmez-Dospinescu-Niziol. Along the way, we describe, in terms of differential forms, the geometric pro-étale cohomology of the positive de Rham period sheaf on any connected, paracompact, smooth rigid-analytic variety over a $p$-adic field, and we do it with coefficients. A key new ingredient is the condensed mathematics recently developed by Clausen-Scholze.

math.NT↗

arXiv2023

Rational $p$-adic Hodge theory for rigid-analytic varieties

We study a cohomology theory for rigid-analytic varieties over $\mathbb{C}_p$, without properness or smoothness assumptions, taking values in filtered quasi-coherent complexes over the Fargues-Fontaine curve, which compares to other rational $p$-adic cohomology theories for rigid-analytic varieties $-$ namely, the rational $p$-adic pro-étale cohomology, the Hyodo-Kato cohomology, and the infinitesimal cohomology over the positive de Rham period ring. In particular, this proves a conjecture of Le Bras. Such comparison results are made possible thanks to the systematic use of the condensed and solid formalisms developed by Clausen-Scholze. As applications, we deduce some general comparison theorems that describe the rational $p$-adic pro-étale cohomology in terms of de Rham data, thereby recovering and extending results of Colmez-Niziol.

math.AG↗