arXiv subjects

Daxin Xu

Publications and source records attributed to Daxin Xu.

10 recordsLinked to original sources

arXiv2026

Frobenius structure on rigid connections and arithmetic applications

We construct the natural Frobenius structures on two families of rigid irregular $\check{G}$-connections on $\mathbb{G}_m$ (or $\mathbb{A}^1$) for a split simple group $\check{G}$: (i) the $θ$-connections arising from Vinberg's $θ$-groups introduced by Chen and Yun; (ii) the Airy connection of Jakob--Kamgarpour--Yi generalizing the classical Airy equations. These data form the $p$-adic companions of the $\ell$-adic local systems introduced by Yun and Jakob--Kamgarpour--Yi. Via the Frobenius structures, we study the local monodromy representations of these local systems at the unique wildly ramified point and verify the prediction of Reeder--Yu on epipelagic Langlands parameters in our setting. We calculate the global geometric monodromy group of a special Airy $\check{G}$-local system via its local monodromy. We show the cohomological rigidity and the physical rigidity of these local systems, as conjectured by Heinloth--Ngô--Yun.

math.NT↗

arXiv2024

$p$-adic non-abelian Hodge theory for curves via moduli stacks

For a smooth projective curve $X$ over $\mathbb C_p$ and any reductive group $G$, we show that the moduli stack of $G$-Higgs bundles on $X$ is a twist of the moduli stack of v-topological $G$-bundles on $X_v$ in a canonical way. We explain how a choice of an exponential trivialises this twist on points. This yields a geometrisation of Faltings' $p$-adic Simpson correspondence for $X$, which we recover as a homeomorphism between the points of moduli spaces. We also show that our twisted isomorphism sends the stack of $p$-adic representations of $π_1(X)$ to an open substack of the stack of semi-stable Higgs bundles of degree $0$.

math.AG↗

arXiv2023

Irregular Hodge filtration of hypergeometric differential equations

Fedorov and Sabbah--Yu calculated the (irregular) Hodge numbers of hypergeometric connections. In this paper, we study the irregular Hodge filtrations on hypergeometric connections defined by rational parameters, and provide a new proof of the aforementioned results. Our approach is based on a geometric interpretation of hypergeometric connections, which enables us to show that certain hypergeometric sums are everywhere ordinary on $|\mathbb{G}_{m,\mathbb{F}_p}|$ (i.e. "Frobenius Newton polygon equals to irregular Hodge polygon").

math.AG↗

arXiv2023

Drinfeld's lemma for $F$-isocrystals, II: Tannakian approach

We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from $\ell$-adic to $p$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.

math.NT↗

arXiv2022

Hypergeometric sheaves for classical groups via geometric Langlands

In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as $\textrm{GL}_n$-local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as $\check{G}$-local systems, for a classical group $\check{G}$. This article aims to realize the geometric Langlands correspondence for these $\check{G}$-local systems. We study this problem from two aspects. In the first approach, we define the hypergeometric automorphic data for a classical group $G$ in the framework of Yun, one of whose local components is a new class of euphotic representations in the sense of Jakob-Yun. We prove the rigidity of hypergeometric automorphic data under natural assumptions, which allows us to define $\check{G}$-local systems $\mathcal{E}_{\check{G}}$ on $\mathbb{G}_m$ as Hecke eigenvalues (in both $\ell$-adic and de Rham setting). In the second approach (which works only in the de Rham setting), we quantize an enhanced ramified Hitchin system, following Beilinson-Drinfeld and Zhu, and identify $\mathcal{E}_{\check{G}}$ with certain $\check{G}$-opers on $\mathbb{G}_m$. Finally, we compare these $\check{G}$-opers with hypergeometric local systems.

math.AG↗

arXiv2022

Parallel transport for Higgs bundles over p-adic curves

Faltings conjectured that under the p-adic Simpson correspondence, finite dimensional p-adic representations of the geometric \'etale fundamental group of a smooth proper p-adic curve X are equivalent to semi-stable Higgs bundles of degree zero over X. In this article, we establish, over a p-adic curve of genus $g\ge 2$, an equivalence between these representations and Higgs bundles, whose underlying bundles potentially admit a strongly semi-stable reduction of degree zero. We show that these Higgs bundles are semi-stable of degree zero and investigate some evidence for the aforementioned conjecture.

math.AG↗

arXiv2019

Bessel $F$-isocrystals for reductive groups

We construct the Frobenius structure on a rigid connection $\mathrm{Be}_{\check{G}}$ on $\mathbb{G}_m$ for a split reductive group $\check{G}$ introduced by Frenkel-Gross. These data form a $\check{G}$-valued overconvergent $F$-isocrystal $\mathrm{Be}_{\check{G}}^{\dagger}$ on $\mathbb{G}_{m,\mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $\check{G}$-local system $\mathrm{Kl}_{\check{G}}$ constructed by Heinloth-Ngô-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $\mathrm{Be}_{\check{G}}^{\dagger}$ when $\check{G}$ is almost simple (which recovers the calculation of monodromy group of $\mathrm{Kl}_{\check{G}}$ due to Katz and Heinloth-Ngô-Yun), and establish functoriality between different Kloosterman $\check{G}$-local systems as conjectured by Heinloth-Ngô-Yun. We show that the Frobenius Newton polygons of $\mathrm{Kl}_{\check{G}}$ are generically ordinary for every $\check{G}$ and are everywhere ordinary on $|\mathbb{G}_{m,\mathbb{F}_p}|$ when $\check{G}$ is classical or $G_2$.

math.AG↗

arXiv2019

Lifting the Cartier transform of Ogus-Vologodsky modulo $p^n$

Let $W$ be the ring of the Witt vectors of a perfect field of characteristic $p$, $\mathfrak{X}$ a smooth formal scheme over $W$, $\mathfrak{X}'$ the base change of $\mathfrak{X}$ by the Frobenius morphism of $W$, $\mathfrak{X}_{2}'$ the reduction modulo $p^{2}$ of $\mathfrak{X}'$ and $X$ the special fiber of $\mathfrak{X}$. We lift the Cartier transform of Ogus-Vologodsky defined by $\mathfrak{X}_{2}'$ modulo $p^{n}$. More precisely, we construct a functor from the category of $p^{n}$-torsion $\mathscr{O}_{\mathfrak{X}'}$-modules with integrable $p$-connection to the category of $p^{n}$-torsion $\mathscr{O}_{\mathfrak{X}}$-modules with integrable connection, each subject to suitable nilpotence conditions. Our construction is based on Oyama's reformulation of the Cartier transform of Ogus-Vologodsky in characteristic $p$. If there exists a lifting $F:\mathfrak{X}\to \mathfrak{X}'$ of the relative Frobenius morphism of $X$, our functor is compatible with a functor constructed by Shiho from $F$. As an application, we give a new interpretation of Faltings' relative Fontaine modules and of the computation of their cohomology.

math.AG↗

arXiv2018

On higher direct images of convergent isocrystals

Let k be a perfect field of characteristic p>0 and W the ring of Witt vectors of k. In this article, we give a new proof of the Frobenius descent for convergent isocrystals on a variety over k relative to W. This proof allows us to deduce an analogue of the de Rham complexes comparaison theorem of Berthelot without assuming a lifting of the Frobenius morphism. As an application, we prove a version of Berthelot's conjecture on the preservation of convergent isocrystals under the higher direct image by a smooth proper morphism of k-varieties.

math.AG↗

arXiv2016

Parallel transport and the p-adic Simpson correspondence

Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful; whether the cohomology of the local systems produced by this functor admits a Hodge-Tate filtration; and whether their construction is compatible with the p-adic Simpson correspondence developed by Faltings. We answer these questions in this article.

math.AG↗