Given a proper smooth $p$-adic variety, we show a comparison theorem for the $p$-adic Simpson correspondence constructed by Faltings and Riemann-Hilbert correspondence constructed by Scholze. As an application we formulate a sufficient condition for $\overline{\mathbb Q}_p$-local system being de Rham. We study a $p$-adic analogue of Simpson's $\mathbb C^*$-action on the set of isomorphism classes of Higgs bundles and the corresponding Galois action on the set of isomorphism classes of generalized representations of the étale fundamental group.
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 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…
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. Following the one we initiated previously, we present an overview of a new monograph developing new features of the p-adic Simpson correspondence, inspired by our construction of the relative Hodge-Tate spectral sequence. First, we address the connection to Hodge-Tate local systems. Second, we establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, we expand the scope of our…
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…
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 …
Let X/k be an abelian variety over an algebraically closed field k of characteristic p > 0. In this paper, using the Azumaya property of the sheaf of crystalline differential operators and the Morita equivalence, we show that etale locally over the Hitchin base, the moduli stack of Higgs bundles on the Frobenius twist X' is equivalent to that of local systems on X. We follow the approach of [Gro16].
The notion of flat $λ$-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for $λ$-flat bundles ($λ\neq 0$), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable $λ$-flat bundles ($λ\neq 0$). On the other hand, we study two applications of the Simpson--Mochizuki correspond…
We define the $p^m$-curvature map on the sheaf of differential operators of level $m$ on a scheme of positive characteristic $p$ as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of $p^m$-curvature on differential modules of level $m$. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level $m$. Then, using a lifting modulo $p^2$ of Frobenius, we are able to define a Frobenius map on differential operators of level $m$ as dual to some divided Frobenius on infinitesimal n…
We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This first article is devoted to the local aspects of the theory.
We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This second article is devoted to the global aspects of the theory.
This work is intended as an introduction to the statement and the construction of the local Langlands correspondence for GL(n) over p-adic fields. The emphasis lies on the statement and the explanation of the correspondence.
We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. The aim of this article is to give an extensive overview of the theory that has been developped in two articles, the first one (arXiv:1102.5466) devoted to the local aspects and the second one (arXiv:1301.0904) to the global aspects.
We construct infinitely many non-isotrivial families of abelian varieties over given four punctured projective lines. These families lead to algebraic solutions of Painleve VI equation. Finally, based on a recent paper by Lin-Sheng-Wang, we prove a complete characterization for the locus of motivic Higgs bundles in the moduli space as fixed points of an ``additive'' self-map. This is a note based on the lecture given by the second named author on 04 Nov. 2022 at Tsinghua University.
We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact Kähler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on Kähler groups proved using the Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of compact Sasakian manifolds.
We show that the standard proof of the Springer correspondence in positive characteristic (via Deligne-Fourier transform) works verbatim in characteristic zero, up to replacing Deligne-Fourier transform by another etale Fourier transform introduced by the author in a previous work. The construction of this Fourier transform uses methods from p-adic analytic geometry.
We construct a functor from the category of p-adic etale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection over its "base change to B_dR", which can be regarded as a first step towards the sought-after p-adic Riemann-Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at ev…
A $p$-adic Riemann--Hilbert correspondence over Robba rings
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院晨兴数学中心）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
For a smooth, projective, complex algebraic variety $X$, the Riemann--Hilbert correspondence establishes a complex analytic isomorphism between the `Betti moduli space' of rank $n$ local systems on $X^\mathrm{an}$ and the `de Rham moduli space' of rank $n$ vector bundles with flat connection on $X$. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are `bi-algebraic' for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We ad…
We describe two candidates for a local p-adic Jacquet-Langlands correspondence and using patching we show that they are in fact isomorphic. We then study locally algebraic vectors of the given correspondence.
For a smooth projective geometrically uniruled threefold defined over a perfect field we show that there exists a canonical abelian variety over the field, namely the second algebraic representative, whose rational Tate modules model canonically the third l-adic cohomology groups of the variety for all primes l. In addition, there exists a rational correspondence inducing these identifications. In the case of a geometrically rationally chain connected variety, one obtains canonical identifications between the integral Tate modules of the second algebraic representative and the third l-adic coh…
We reprove the Local Langlands Correspondence for $\GL_n$ over $p$-adic fields as well as the existence of $\ell$-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor. In contrast to the proofs of the Local Langlands Correspondence given by Henniart and Harris-Taylor our proof completely by-passes the numerical Local Langlands Correspondence of Henniart. Instead, we make use of a previous result describing the inertia-invariant nearby c…
The nonabelian Hodge correspondence (Corlette-Simpson correspondence), between the polystable Higgs bundles with vanishing Chern classes on a compact Kähler manifold $X$ and the completely reducible flat connections on $X$, is extended to the Fujiki class $\mathcal C$ manifolds.
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…
We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze's basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results. Our key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the …
On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a $p$-adic analogue of Deligne's classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of resi…
Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come from families of abelian varieties. When $X$ is a projective variety, we prove a Lefschetz-style theorem for abelian schemes of $\text{GL}_2$-type on $X$, modeled after a theorem of Simpson. If one assumes a strong form of Deligne's ($p$-adic) \emph{companions conjecture} from Weil II, this implies that our conjecture for projective varieties also reduces to the …
We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge-Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proo…
Constructing families of abelian varieties of $\text{GL}_2$-type over $4$-punctured complex projective line via $p$-adic Hodge theory and Langlands correspondence and application to algebraic solutions of Painleve VI equation
In this article we establish some constraints on the eigenvalues for the action of a self map of a proper variety on its $\ell$-adic cohomology. The essential ingredients are a trace formula due to Fujiwara, and the theory of weights.
In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that n is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic techn…
A stacky $p$-adic Riemann--Hilbert correspondence on Hitchin-small locus
Peking University（北京大学）；University of Copenhagen（哥本哈根大学）；Academy of Mathematics and Systems Science, University of Chinese Academy of Science（中国科学院数学与系统科学研究院）；Princeton University（普林斯顿大学）；Fudan University（复旦大学）
