Uniformization of <i>p</i>-adic curves via Higgs–de Rham flows

Lan, Guitang; Sheng, Mao; Yang, Yueting; Zuo, Kang

doi:10.1515/crelle-2016-0020

Cited by 8 publications

(6 citation statements)

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“…By working on this problem for a very simple kind of rank two Higgs bundles of degree zero (the so-called Higgs bundle with maximal Higgs field) over a curve, we have found a p-adic analogue of the Hitchin-Simpson's uniformization of hyperbolic curves which relates intimately the above theory to the theory of ordinary curves due to S. Mochizuki ([23]). In particular, the canonical lifting theorem of Mochizuki for ordinary curves has been basically recovered in our recent work [16].…”

Section: Riemann-hilbert Correspondencementioning

confidence: 77%

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

2019

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Let k be the algebraic closure of a finite field of odd characteristic p and X a smooth projective scheme over the Witt ring W (k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow 1 and prove that the category of periodic Higgs-de Rham flows over X/W (k) is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of theétale fundamental group π 1 (X K ) of the generic fiber of X, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bundle over the special fiber X k of X of rank ≤ p initiates a semistable Higgs-de Rham flow and thus those of rank ≤ p − 1 with trivial Chern classes induce k-representations of π 1 (X K ). A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic p, it was constructed by Ogus-Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus-Vologodsky correspondence of Shiho. 45 7. Rigidity theorem for Fontaine modules 51 Appendix A. Semistable Higgs bundles of small ranks are strongly Higgs semistable 55 References 59

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Section: Riemann-hilbert Correspondencementioning

confidence: 77%

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

2019

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“…1 also has the logarithmic version. When the log structure is given by a simple normal crossing divisor, an explicit construction of the log inverse Cartier functor is given in the Appendix of [18].…”

Section: Twisted Fontaine-faltings Modulesmentioning

confidence: 99%

“…. By the same argument used in the proof of Proposition 1.4 in [18], we are going to show that ρ ′ is strongly irreducible. Hence ρ is strongly irreducible.…”

mentioning

confidence: 98%

“…We recall the notion of strong irreducibility of representations, which is introduced in [18], Proposition 1.4. Let ρ : π ét 1 (X 0 K ) → PGL r (Z ur p ) be a representation and ρ be the restriction of ρ to the geometric fundamental group π ét 1 (X 0 K ).…”

mentioning

confidence: 99%

“…Cp is nonzero, (f * O(−1), 0) Cp in the above short exact sequence is the unique rank-1 Higgs subbundle. Hence, one obtains 2 The original inverse Cartier transform is defined by Ogus and Vologodsky [28] for characteristic p. And lately, it was generalized to the truncated version by Lan-Sheng-Zuo [19] and to the logarithmic version by Schepler [30] and Lan-Sheng-Yang-Zuo [18]. Here we need a truncated logarithmic version.…”

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confidence: 99%

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Projective Crystalline Representations of Étale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

Sun¹,

Yang²,

Zuo³

2017

Preprint

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This paper contains three new results. 1.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of étale fundamental groups introduced in [7, 10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. 2.We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. 3. We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on P 1 with logarithmic structure on marked points D := {x1, ..., xn} for n ≥ 4 and construct infinitely many geometrically absolutely irreducible PGL2(Z ur p )-crystalline representations of π et 1 (P 1We find an explicit formula of the selfmap for the case {0, 1, ∞, λ} and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve C λ defined by y 2 = x(x − 1)(x − λ) with the order coprime to p.2.2. The category of projective representations 2.3. Gluing representations and projective representations 2.4. Comparing representations associated to local Fontaine-Faltings modules underlying isomorphic filtered de Rham sheaves 2.5. The functor D P 3. Twisted periodic Higgs-de Rham flows 3.1. Higgs-de Rham flow over X n ⊂ X n+1 3.2. Twisted periodic Higgs-de Rham flow and equivalent categories 3.3. Twisted Higgs-de Rham self map on moduli schemes of semi-stable Higgs bundle with trivial discriminant 3.4. Sub-representations and sub periodic Higgs-de Rham flows 4. Constructing crystalline representations of étale fundamental groups of p-adic curves via Higgs bundles 4.1. Connected components of the moduli space M 4.2. Self-maps on moduli spaces of Higgs bundles on P 1 with marked points 4.3. An explicit formula of the self-map in the case of four marked points. 4.4. Lifting of twisted periodic logarithmic Higgs-de Rham flow on the projective line with marked points and strong irreducibility 4.5. Examples of dynamics of Higgs-de Rham flow on P 1 with four-marked points 4.6. Question on periodic Higgs bundles and torsion points on the associated elliptic curve. 4.7. Question on ℓ-adic representation and ℓ-to-p companions. 4.8. Projective F -units crystal on smooth projective curves 5. Base change of twisted Fontaine-Faltings modules and twisted Higgs-de Rham flows over very ramified valuation rings 5.1. Notations in the case of Spec k. 5.2. Base change in the small affine case. 5.3.Categories and Functors on proper smooth variety over very ramified valuation ring W π 5.4. degree and slope 6. Appendix: explicit formulas 6.1. Self-map 6.2. Multiple by p map on ellipt...

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A Note on the Filtered Decomposition Theorem

Zhang

2022

Commun. Math. Stat.

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Uniformization of p-adic curves via Higgs–de Rham flows

Cited by 8 publications

References 13 publications

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

Projective Crystalline Representations of Étale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow

A Note on the Filtered Decomposition Theorem

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