arXiv subjects

Arthur-César Le Bras

Publications and source records attributed to Arthur-César Le Bras.

14 recordsLinked to original sources

arXiv2026

Rational analytic prismatization

We develop a general theory of analytic prismatization with $\mathbb{Q}_p$-coefficients in rigid analytic geometry over $\mathbb{Q}_p$. We prove an equivalence of perfect complexes on the perfect analytic prismatization modulo Frobenius with perfect complexes on the relative Fargues--Fontaine curve for any smooth partially proper rigid analytic variety. In particular, we explain how this equivalence conceptualizes the decompletion techniques by passing to locally analytic vectors which appear in (geometric) Sen theory

math.AG↗

arXiv2026

A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve

We develop a 6-functor formalism $\mathcal{D}_{[0,\infty)}(-)$ with $\mathbb{Z}_p$-linear coefficients on small v-stacks, and discuss consequences for duality and finiteness for pro-étale cohomology of rigid-analytic varieties of general pro-étale $\mathbb{Q}_p$-local systems as well as first examples motivated by a potential $p$-adic analog of Fargues-Scholze's geometrization program of the local Langlands correspondence.

math.AG↗

arXiv2026

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.

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arXiv2025

v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack

We describe the category of continuous semilinear representations and their cohomology for the Galois group of a $p$-adic field $K$ with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring $B_{\mathrm{Sen}}$ in a geometric way.

math.NT↗

arXiv2025

Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces

Let $X$ be a quasi-compact quasi-separated $p$-adic formal scheme that is smooth either over a perfectoid $\mathbb{Z}_p$-algebra or over some ring of integers of a $p$-adic field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of $X$ up to isogeny to perfect complexes on the v-site of the generic fibre of $X$. Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived $p$-adic Simpson functor.

math.AG↗

arXiv2023

The small $p$-adic Simpson correspondence in terms of moduli spaces

For any rigid space over a perfectoid extension of $\mathbb Q_p$ that admits a liftable smooth formal model, we construct an isomorphism between the moduli stacks of Hitchin-small Higgs bundles and Hitchin-small v-vector bundles. This constitutes a moduli-theoretic improvement of the small $p$-adic Simpson correspondence of Faltings, Abbes-Gros, Tsuji and Wang. Our construction is based on the Hodge-Tate stack of Bhatt-Lurie. We also prove an analogous correspondence in the arithmetic setting of rigid spaces of good reduction over $p$-adic fields.

math.AG↗

arXiv2023

The de Rham-Fargues-Fontaine cohomology

We show how to attach to any rigid analytic variety $V$ over a perfectoid space $P$ a rigid analytic motive over the Fargues-Fontaine curve $\mathcal{X}(P)$ functorially in $V$ and $P$. We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasi-coherent sheaves over $\mathcal{X}(P)$, and we show that its cohomology groups are vector bundles if $V$ is smooth and proper over $P$ or if $V$ is quasi-compact and $P$ is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit $\mathbb{B}^1$-homotopies, the motivic proper base change and the formalism of solid quasi-coherent sheaves.

math.AG↗

arXiv2022

Prismatic Dieudonné theory

We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $\mathrm{DM}^{\rm adm}(R)$ of \textit{admissible prismatic Dieudonné crystals over $R$} and a natural functor from $p$-divisible groups over $R$ to $\mathrm{DM}^{\rm adm}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.

math.AG↗

arXiv2021

A Fourier Transform for Banach-Colmez spaces

We define and study an $\ell$-adic Fourier transform for a relative version of Banach-Colmez spaces (over a perfectoid space which is not necessarily a geometric point), which can be thought of as some analytic analogue of the $\ell$-adic Fourier transform for unipotent perfect group schemes. We explicitly describe it on some examples.

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arXiv2021

Averaging functors in Fargues' program for GL_n

We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues' program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm{GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm{GL}_2$ are irreducible. We also attach to any irreducible $\ell$-adic Weil representation of degree $n$ an Hecke eigensheaf on $\mathrm{Bun}_n$, and show, using the local Langlands correspondence and recent results of Hansen and Kaletha-Weinstein, that it satisfies most of the requirements of Fargues' conjecture for $\mathrm{GL}_n$.

math.NT↗

arXiv2021

The $p$-completed cyclotomic trace in degree $2$

We prove that for a quasi-regular semiperfectoid $\mathbb{Z}_p^{\rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;\mathbb{Z}_p)$ of $R$ to the topological cyclic homology $\mathrm{TC}(R;\mathbb{Z}_p)$ of $R$ identifies on $π_2$ with a $q$-deformation of the logarithm.

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arXiv2018

Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve

We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.

math.NT↗

arXiv2018

Espaces de Banach-Colmez et faisceaux cohérents sur la courbe de Fargues-Fontaine

We give a new definition, simpler but equivalent, of the abelian category of Banach-Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues-Fontaine curve. One goes from one category to the other by changing the t-structure on the derived category. Along the way, we obtain a description of the pro-étale cohomology of the open disk and the affine space, of independent interest.

math.NT↗

arXiv2017

Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique

We describe the de Rham complex of the étale coverings of Drinfeld's p-adic upper half-plane for GL_2(Q_p). Conjectured by Breuil and Strauch, this description gives a geometric realization of the p-adic local Langlands correspondence for certain two-dimensional de Rham representations of the absolute Galois group of Q_p.

math.NT↗