Serin Hong
Given two arbitrary vector bundles on the Fargues-Fontaine curve, we give an explicit criterion in terms of Harder-Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely combinatorial and builds upon the dimension analysis of certain moduli spaces of bundle maps developed in [BFH+22].
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We extend our previous theory of etale parallel transport to a larger class of slope zero vector bundles on p-adic curves. The new class is stable under pullback by ramified coverings. We also construct p-adic representations of a central extension of the fundamental group for certain bundles of non-zero slope.
We define and study a certain category of vector bundles on a p-adic curve to which we can associate in a functorial way finite dimensional p-adic representations of the geometric fundamental group. Among other things we investigate two different relations of these constructions with the Hodge-Tate decomposition.
For a vector bundle E on a model of a smooth projective curve over a p-adic number field a p-adic representation of the geometric fundamental group of X has been defined in work with Annette Werner if the reduction of E is strongly semistable of degree zero. In the present note we calculate the reduction of this representation using the theory of Nori's fundamental group scheme.
We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.
We show how to functorially attach continuous $p$-adic representations of the profinite fundamental group to vector bundles with numerically flat reduction on a proper rigid analytic variety over $\mathbb{C}_p$. This generalizes results by Deninger and Werner for vector bundles on smooth algebraic varieties. Our approach uses fundamental results on the pro-étale site of a rigid analytic variety introduced by Scholze. This enables us to get rid of the smoothness condition and to work in the analytic category. Moreover, under some mild conditions, the functor we construct gives a full embedding …
We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of deg…
We use Scholze's framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the étale, pro-étale, and the $v$-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb{C}_p$ this enables us to prove a category equivalence between $\mathbb{C}_p$-local systems with integral…
Parallel transport for Higgs bundles over p-adic curves
本文在亏格g≥2的p进曲线上，建立了几何平展基本群的有限维p进表示与具强半稳定约化的希格斯丛的等价关系，为法尔廷斯的p进辛普森对应猜想提供佐证。
We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a $p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $p$-adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $p$-adic analog of higher-dimensional generalizations of the classical …
In [DW05] and [DW07], C. Deninger and A. Werner developed a partial p-adic analogue of the classical Narasimhan-Seshadri correspondence between vector bundles and representations of the fundamental group. We will investigate the various monodromy groups that occur in this theory, that is the image of these representations and their Zariski closure as well as the Tannaka dual group of these vector bundles.
We prove that a category of degree zero vector bundles with "potentially strongly semistable reduction" on a p-adic curve is a neutral Tannakian category. We also make a first study of the corresponding affine group scheme. In particular, we determine its group of connected components using a theorem of Weil.
A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gauß--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein--Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable $p$-adic Eisenstein measure through $p$-adic theta functions of the Poincaré b…
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamen…
We define functorial isomorphisms of parallel transport along étale paths for a class of principal $G$-bundles on a $p$-adic curve. Here $G$ is a connected reductive algebraic group of finite presentation and the considered principal bundles are just those with potentially strongly semistable reduction of degree zero. The constructed isomorphisms yield continous functors from the étale fundamental groupoid of the given curve to the category of topological spaces with a simply transitive continous right $G(\mathbb{C}_{p})$-action. This generalizes a construction in the case of vector bundles on…
The p-adic Simpson correspondence due to Faltings is a p-adic analogue of non-abelian Hodge theory. The following is the main result of this article: The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions. In the complex setting, Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties. We give a p-adic analogue of Simpson's result.
For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles of higher rank. We also compare this construction to the theory of parallel transport for vector bundles on p-adic curves developed in mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.
The aim of the present paper is to provide a new aspect of the $p$-adic Teichmüller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichmüller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in…
$\ell$-adic local systems and Higgs bundles: the generic case
本文证明有限域曲线上具有一般分歧的 $\ell$-adic 局部系统（自动不可约）的 Frobenius 不动点计数，等于稳定对数 Higgs 丛（分裂情形为抛物 Higgs 丛）的计数，相差 $q$ 的幂次。
We study the motivic and $\ell$-adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D.~Orlov and J.~Burke - M.~Walker to give a formula for the $\ell$-adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension $n$.
We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the ordinary locus of the modular curve in terms of certain $p$-adic analytic moment functions associated to Katz' two-variable $p$-adic Eisenstein measure. The present work generalizes previous results of Bannai-Kobayashi-Tsuji and Bannai-Kings on the syntomic Eisenstein classes.
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 …
For an abeloid variety $A$ over a complete algebraically closed field extension $K$ of $\mathbb Q_p$, we construct a $p$-adic Corlette-Simpson correspondence, namely an equivalence between finite-dimensional continuous $K$-linear representations of the Tate module and a certain subcategory of the Higgs bundles on $A$. To do so, our central object of study is the category of vector bundles for the $v$-topology on the diamond associated to $A$. We prove that any pro-finite-étale $v$-vector bundle can be built from pro-finite-étale $v$-line bundles and unipotent $v$-bundles. To describe the latte…
Lift-independence problem in the $p$-adic Simpson correspondence for curves
研究曲线情形下 $p$-进 Simpson 对应中 Higgs 丛与 $v$-丛的等价性是否依赖于提升的选择，证明了小秩半稳定提升独立 Higgs 丛的 Higgs 场为零，并构造了非零 Higgs 场的半稳定提升独立 Higgs 丛。
$p$-adic Simpson correspondences for principal bundles in abelian settings
本文研究刚性空间上主丛的 $p$-进 Simpson 对应，证明交换群情形由李群对数满射性刻画，并将 Faltings 小对应推广至普通阿贝尔簇上的一般刚性群。
Let $p$ be a prime. This papers aims at investigating sheaf cohomology of a broader class of $p$-adic period domains, other then the Drinfeld's upper half space (cf. \cite{O2}). Concretely, we let $\mathbf{G}$ be a split connected reductive group over $\mathbb{Q}_p$ and restrict our attention to the $p$-adic period domain $\mathscr{F}^{\mathrm{wa}}$ which parametrizes the weakly admissible filtrations on the trivial $\mathbf{G}$-isocrystal inside a complete flag variety $\mathscr{F}$. Then, we consider sheaf cohomology of $\mathscr{F}^{\mathrm{wa}}$ with coefficients in vector bundles which ar…
Mirror symmetry for parabolic Higgs bundles via $p$-adic integration
该研究运用$p$-进积分技术，基于非抛物情形的前期工作，证明了抛物型希格斯丛模空间的Hausel-Thaddeus拓扑镜像对称猜想，还证明了抛物型$\text{GL}_n$-希格斯丛光滑模空间的$E$-多项式与底层向量丛次数无关。
p-adic Borel extension for local Shimura varieties
本文证明了局部Shimura品种满足p-进刚性解析的Borel扩展定理，并扩展了Rapoport-Zink空间的相关结果。
The small $p$-adic Simpson correspondence in the semi-stable reduction case
本文将光滑形式概型上的小 Simpson 对应已知结果推广至半稳定情形，为可提升半稳定形式概型建立了 Hitchin-small integral $v$-bundles 与 Higgs bundles 范畴及对应模栈的等价关系。
We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme.
Chern classes of automorphic vector bundles, II
本文证明Hodge型Shimura簇环面紧化上自守向量丛典范扩张的ℓ-adic Chern类可下降至最小紧化上同调，且在Galois群作用下不变。
Rank 2 $\ell$-adic local systems and Higgs bundles over a curve
针对有限域上曲线补集的秩2 ℓ-adic局部系统，研究被Frobenius迭代固定的给定tame局部单模的系统数量，证实Deligne猜想并将结果用Higgs丛数量表示。
We introduce a notion of inertial equivalence for integral $\ell$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $\ell$-adic representations of the absolute Galois group of a global field lifting a fixed absolutely irreducible residual representation and with given inertial type outside a fixed finite set of places is uniformly bounded independent of the inertial type.
Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation
本文证明任意全虚数域上代数Hecke特征标的临界$L$-值除以周期为代数整数，并构造$p$-进测度插值普通情形下的临界值，推广了Damerell、Shimura和Katz的结果。
The section conjecture for the toric fundamental group over $p$-adic fields
本文在 $p$-adic 域上证明了环面基本群的 Grothendieck 截面猜想，并将 Selmer 截面重新解释为环面 Galois 截面。
Using the new approach to analytic geometry developed by Clausen and Scholze by means of condensed mathematics, we prove that for every affinoid analytic adic space $X$, pseudocoherent complexes, perfect complexes, and finite projective modules over $\mathcal{O}_X(X)$ form a stack with respect to the analytic topology on $X$; in particular, we prove that the category of vector bundles on $X$ is equivalent to the category of finite projective modules over $\mathcal{O}_X(X)$. To that end, we construct a fully faithful functor from the category of complete Huber pairs to the category of analytic …
We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's conjectures which predict that the Chern classes in the Chow groups vanish rationally.
For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb Q_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: The $p$-adic character variety is canonically an étale twist of the moduli space of topological torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$-line bundles: We develop a theory of topological torsion subsheaves of $v$-sheaves and apply this to the diamantine $v$-Pica…
Given an automorphic line bundle ${\mathcal O}_X(k)$ of weight $k$ on the Drinfel'd upper half plane $X$ over a local field $K$, we construct a ${\rm GL}_2(K)$-equivariant integral lattice ${\mathcal O}_{\widehat{\mathfrak X}}(k)$ in ${\mathcal O}_X(k)\otimes_K\widehat{K}$, as a coherent sheaf on the formal model $\widehat{\mathfrak{X}}$ underlying $X\otimes_K\widehat{K}$. Here $\widehat{K}/K$ is ramified of degree $2$. This generalizes a construction of Teitelbaum from the case of even weight $k$ to arbitrary integer weight $k$. We compute $H^*(\widetilde{\mathfrak{X}},{\mathcal O}_{\widehat{…
Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the curve to the ring $W(k)$ of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over $X/W$. As a consequence, it gives rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group $π_1(X_K)$ of the generic fiber of $X$. This curve $X$ and its associated representation is in close relation wit…
On a p-adic version of Narasimhan and Seshadri's theorem
本文研究 Faltings 关于 p 进 Simpson 对应像的提问，证明在好约化及 p 足够大的条件下，潜在强半稳定约化等价于强半稳定约化，从而给出否定回答。
A $p$-adic Simpson correspondence for singular rigid-analytic varieties
研究p进Simpson对应，将正则解析簇上的pro-étale向量束与Higgs束联系，推广Faltings和Heuer的工作至任意正则解析簇。
We study the ramification of fierce cyclic Galois extensions of a local field $K$ of characteristic zero with a one-dimensional residue field of characteristic $p>0$. Using Kato's theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs $(δ_i,ω_i)$, where $δ_i$ is a positive rational number and $ω_i$ a differential form on the residue field of $K$. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of $K$.
The $p$-adic monodromy theorem over algebraic-affinoid algebras
本文研究代数仿射代数参数化的Fargues-Fontaine曲线上$G_K$-等变向量丛族的$p$-进霍奇理论性质，定义相关表示概念并证明$p$-进单值化定理，推广了Berger-Colmez的工作，还在去掉自由性条件下给出了等变线丛族的分类。
Faltings extension and Hodge-Tate filtration for abelian varieties over $p$-adic local fields with imperfect residue fields
针对剩余域不完备的特征0p进局部域，定义Z_p上O_K的Faltings扩张，推广Fontaine1981年的构造，为该类域上的阿贝尔簇构建Hodge-Tate滤过。
In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product (of equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [V], and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety…
A dormant indigenous bundle is an integrable projective line bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in the $p$-adic Teichmuller theory developed by S. Mochizuki. Kirti Joshi proposed a conjecture concerning an explicit formula for the degree over the moduli stack of curves of the moduli stack classifying dormant indigenous bundles. In this paper, we give a proof for this conjecture of Joshi.
Non-liftability of Families of Abelian Varieties with Small $l$-adic Local System
本文研究特征 $p$ 上光滑正则曲线上的阿贝尔簇族，证明其霍奇包非 nef 且无法提升到 $W_2(k)$，并建立一个阿拉凯洛夫型不等式。
We define functorial isomorphisms of parallel transport along etale paths for a class of G-principal bundles on a p-adic curve where G is a connected reductive algebraic group of finite presentation. This class consists of all principal bundles with potentially strongly semistable reduction of degree zero. In particular, this construction gives us a continous functor from the etale fundamental grupoid of the given curve to the category of topological spaces with a simply transitive continous right G(\mathbb{C}_{p})-action for every such principal bundle. This generalizes the construction of fu…
$p$-adic non-abelian Hodge theory for curves via moduli stacks
本文通过模堆的典范扭转，将曲线上的 $p$-adic 非阿贝尔 Hodge 理论几何化，并恢复 Faltings 对应为同胚，同时将表示堆映射到半稳定 Higgs 丛的开子堆。
Canonical extensions of $p$-adic shtukas on toroidal compactifications of Shimura varieties
本文在 Abel 型 Shimura 簇的环面紧化的整模型上，通过研究边界处 shtuka 的“退化”，构造了具有拟抛物水平结构的 $p$-adic shtuka 的典范扩张，并定义了 Pappas-Rapoport 意义下的典范整模型。
The small $p$-adic Simpson correspondence in terms of moduli spaces
本研究针对$\text{Q}_p$的perfectoid扩张上的刚性空间，基于Bhatt-Lurie的Hodge-Tate栈构造两类模栈的同构，改进了小$p$进Simpson对应，并在算术设定下证明了类似结论。
Algorithms for hyperelliptic Mumford Curves $p$-adic Uniformization, $p$-adic integrals and $p$-adic heights
针对超椭圆Mumford曲线，提出计算$p$-adic Schottky群的算法，并利用该均匀化计算$p$-adic Abelian积分和Schneider高度，通过SageMath数值示例验证。
Extensions of vector bundles on the Fargues-Fontaine curve II
该研究针对Fargues-Fontaine曲线上的向量丛扩张问题，完成了由两个任意向量丛生成的所有扩张向量丛的完全分类工作。
I prove the surjectivity of the $p$-adic regulator from Quillen's $K_2$ of Tate curve to the $p$-adic etale cohomology group when the base field is contained in a cyclotomic extension of $Q_p$. This implies the finiteness of torsion part of $K_1$ of Tate curves thanks to Suslin's exact sequence.
Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even…
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 …
Non-archimedean topological mirror symmetry for $SL_n$ and $PGL_n$ Higgs bundles
本文将非阿基米德拓扑镜像对称方法推广到非互素情形，证明任意次数下 $SL_n$ 与 $PGL_n$ Higgs 丛模空间的 gerbe 扭曲 $p$-进体积相等，并建立其与相交上同调及 $BPS$ 上同调的联系。
A $p$-adic Simpson correspondence for smooth proper rigid varieties
本文为光滑真刚性解析空间构造了 $p$-adic Simpson 对应，建立了向量丛与 Higgs 丛之间的范畴等价，推广了 Faltings 的结果到高维及刚性解析情形，方法基于谱簇上 pro-étale 可逆层的模空间研究。
Moduli spaces in $p$-adic non-abelian Hodge theory
本文提出基于模叠比较的 $p$-adic Simpson 对应新方法，引入光滑类空间并构造层化非阿贝尔 Hodge 对应，推广 Faltings 结果，推导 $v$-下降准则，并在 Betti 侧构造 Hitchin 态射类比，给出更几何的猜想性重述。
A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here we show that rationally connected varieties over the maximally unramified extension of the p-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result saying that the p-adics are usually $C_{2}$ …
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