arXiv subjects

Jilong Tong

Publications and source records attributed to Jilong Tong.

8 recordsLinked to original sources

arXiv2024

On the weak Harder-Narasimhan stratification on $B_{\mathrm{dR}}^+$-affine Grassmannian

We consider the Harder-Narasimhan formalism on the category of normed isocrystals and show that the Harder-Narasimhan filtration is compatible with tensor products which generalizes a result of Cornut. As an application of this result, we are able to define a (weak) Harder-Narasimhan stratification on the $B_{\mathrm{dR}}^+$-affine Grassmannian for arbitrary $(G, b, μ)$. When $μ$ is minuscule, it corresponds to the Harder-Narasimhan stratification on the flag varieties defined by Dat-Orlik-Rapoport. And when $b$ is basic, it's studied by Nguyen-Viehmann and Shen. We study the basic geometric properties of the Harder-Narasimhan stratification, such as non-emptiness, dimension and its relation with other stratifications.

math.AG↗

arXiv2022

Weakly admissible locus and Newton stratification in p-adic Hodge theory

The basic admissible locus $\mathcal{F}(G, μ, b)^a$ inside the flag variety $\mathcal{F}(G, μ)$, attached to a reductive group $G$ with a minuscule cocharacter $μ$ of $G$, is a $p$-adic analogue of the complex analytic period spaces. It has an algebraic approximation $\mathcal{F}(G, μ, b)^{wa}$ inside the flag variety, called the weakly admissible locus. On the flag variety $\mathcal{F}(G, μ)$, we have the Newton stratification which has the admissible locus as its unique open stratum. In this paper, we study the relation between the Newton strata and the weakly admissible locus. We show that $\mathcal{F}(G, μ, b)^{wa}$ is maximal (in the sense that it's a union of Newton strata) is equivalent to $(G, μ)$ weakly fully HN-decomposable, it's also equivalent to the condition that the Newton stratification is finer than the Harder-Narasimhan stratification. These equivalent conditions are generalizations of the fully HN-decomposable condition and the weakly accessible condition. Moreover, we give a criterion to determine whether a Newton stratum is completely contained in the weakly admissible locus involving $G$-bundles as extensions of $M$-bundles over the Fargues-Fontaine curve, where $M$ is a Levi subgroup of $G$. When $G=\mathrm{GL}_n$, we also give a combinatorial inductive criterion to determine whether a vector bundle over the Fargues-Fontaine curve is an extension of two given vector bundles.

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arXiv2020

A note on Higgs-de Rham flows of level zero

The notion of Higgs-de Rham flows was introduced by Lan-Sheng-Zuo, as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory. In this short note we investigate a small part of this theory, and study those Higgs-de Rham flows which are of level zero. We improve the original definition of level-zero Higgs-de Rham flows (which works for general levels), and establish a Hitchin-Simpson-type correspondence between such objects and certain representations of fundamental groups in positive characteristic, which generalizes the classical results of Katz. We compare the deformation theories of two sides in the correspondence, and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.

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arXiv2019

Crystalline comparison isomorphisms in $p$-adic Hodge theory: the absolutely unramified case

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-étale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-étale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

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arXiv2013

Néron models of algebraic curves

Let S be a Dedekind scheme with field of functions K. We show that if X_K is a smooth connected proper curve of positive genus over K, then it admits a Néron model over S, i.e., a smooth separated model of finite type satisfying the usual Néron mapping property. It is given by the smooth locus of the minimal proper regular model of X_K over S, as in the case of elliptic curves. When S is excellent, a similar result holds for connected smooth affine curves different from the affine line, with locally finite type Néron models.

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arXiv2013

On torsors under elliptic curves and Serre's pro-algebraic structures

Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the Néron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres. We show that $q$ is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties.

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arXiv2010

Diviseur Theta et Formes Differentielles

In this papers, we study the geometric and arithmetic properties of the theta divisor associated to the sheaf of locally exact differential forms over a curve in positive characteristic. In this published version, we prove a stronger version of the main result of chapiter 5.

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arXiv2010

Etude locale des torseurs sous une courbe elliptique

This article concerns the geometry of torsors under an elliptic curve. Let $\OO_K$ be a complete discrete valuation ring with algebraically closed residue field and function field $K$. Let $π$ be a generator of the maximal ideal of $\OO_K$, and $S=\mathrm{Spec}(\OO_K)$. Suppose that we are given $J_K$ an elliptic curve over $K$, with $J$ the connected component of the $S$-N?ron model of $J_K$. Given $X_K/K$ a torsor of order $d$ under $J_K$, let $X$ be the $S$-minimal regular proper model. Then there is an invertible id?al $\mathcal{I}\subset \OO_K$ such that $\mathcal{I}^{d}=π\OO_X\subset \OO_X$. Moreover, there exists a canonical morphism $q:\Pic^{\circ}_{X/S}\rightarrow J$ which induces a surjective map $q(S):\Pic^{\circ}(X)\rightarrow J(S)$. The purpose of the article is to prove this last morphism $q(S)$ is compatible with respect to the $\mathcal{I}$-adic filtration on $\Pic^{\circ}(X)$, and the $π$-adic filtration on $J(S)$. As a byproduct, we obtain {\textquotedblleft Herbrand functions\textquotedblright}, similar to those Serre used in his description of local class fields (\cite{Serre})

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