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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2019, Issue 747

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

Guitang Lan
/ Mao Sheng
/ Yanhong Yang
/ Kang Zuo
Published Online: 2016-07-12 | DOI: https://doi.org/10.1515/crelle-2016-0020

## Abstract

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 $W\left(k\right)$ of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over $X/W\left(k\right)$. These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group ${\pi }_{1}\left({X}_{K}\right)$ of the generic fiber of X. This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p-adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin–Simpson’s uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.

## References

• [1]

P. Deligne and L. Illusie, Relèvements modulo ${p}^{2}$ et decomposition du complexe de de Rham, Invent. Math. 89 (1987), 247–270. Google Scholar

• [2]

C. Deninger and A. Werner, Vector bundles on p-adic curves and parallel transport, Ann. Sci. Éc. Norm. Supér. 38 (2005), 553–597.

• [3]

H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhäuser, Basel, 1992. Google Scholar

• [4]

G. Faltings, Crystalline cohomology and p-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore 1988), Johns Hopkins University Press, Baltimore (1989), 25–80. Google Scholar

• [5]

• [6]

N. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3) 55 (1987), 59–126. Google Scholar

• [7]

K. Kato, Logarithmic structures of Fontaine–Illusie, Algebraic analysis, geometry, and number theory (Baltimore 1988), Johns Hopkins University Press, Baltimore (1989), 191–224. Google Scholar

• [8]

N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math. Inst. Hautes Études Sci. 39 (1970), 175–232.

• [9]

G.-T. Lan, M. Sheng and K. Zuo, Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups, preprint (2014), http://arxiv.org/abs/1311.6424v2.

• [10]

G.-T. Lan, M. Sheng and K. Zuo, Nonabelian Hodge theory in positive characteristic via exponential twisting, Math. Res. Lett. 22 (2015), no. 3, 859–879.

• [11]

A. Langer, Moduli spaces of sheaves and principal G-bundles, Algebraic geometry. Part 1 (Seattle 2005), Proc. Sympos. Pure Math. 80, American Mathematical Society, Providence (2009), 273–308. Google Scholar

• [12]

S. Mochizuki, A theory of ordinary p-adic curves, Publ. Res. Inst. Math. Sci. 32 (1996), no. 6, 957–1152.

• [13]

A. Ogus, F-crystals, Griffiths transversality, and the Hodge decomposition, Astérisque 221, Société Mathématique de France, Paris, 1994. Google Scholar

• [14]

A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic p, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1–138.

• [15]

D. Schepler, Logarithmic nonabelian Hodge theory in characteristic p, preprint (2008), http://arxiv.org/abs/0802.1977.

• [16]

C. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), no. 3, 713–770.

• [17]

J. Xia, On the deformation of a Barsotti-Tate group over a curve, preprint (2013), http://arxiv.org/abs/1303.2954.

Revised: 2016-03-16

Published Online: 2016-07-12

Published in Print: 2019-02-01

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: SFB/TR 45 “Periods

Award identifier / Grant number: Moduli Spaces and Arithmetic of Algebraic Varieties”

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11471298

Award identifier / Grant number: 11526212

This work is supported by the SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG. The second named author is supported by National Natural Science Foundation of China (Grant No. 11471298, No. 11526212).

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 747, Pages 63–108, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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