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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

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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


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 X1 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(k). These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group π1(XK) 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.


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About the article

Received: 2014-08-03

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, DOI: https://doi.org/10.1515/crelle-2016-0020.

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