# Does anybody do $p$-adic Teichmüller theory?

In "Foundations of $$p$$-adic Teichmüller theory", Mochizuki describes a theory one of whose goals (according to the author) is to generalize Fuchsian uniformization of Riemann surfaces to the $$p$$-adic context. A quick glance through this book reveals a lot of new concepts (with fancy names!) introduced by Mochizuki.

The question is: what are some landmark papers in this theory (after this textbook)? Did anybody other than Mochizuki and his postdocs/students make contributions to this theory? A less mathematical question: are there any people in Western Europe/U.S. working on this topic in the present?

For example, while I think that Jakob Stix is doing some anabelian geometry (and Mochizuki has made major contributions to it), I am not sure if any of his work is specifically building upon "Foundations of $$p$$-adic Teichmüller theory".

P.S. Not being an inter-universalist, I do not know whether "Foundations of $$p$$-adic Teichmüller theory" have anything to do with the IUT. The question, however, is not about IUT. To avoid off-topic debate, let us pretend in this thread that IUT papers do not exist.

• I know nothing about the p-adic case (only the classical theory of complex projective structures on Riemann surfaces), but Mathsci net gives a list of 32 papers referring to Mochizuki's book (and not all of these are by Mochizuki himself). Apr 20, 2019 at 13:44
• What did you get wham you looked up who cited the book on MathSciNet, Zentralblatt, and Google Scholar? That's the obvious thing to do.
– Dirk
Apr 20, 2019 at 15:57
• @Dirk well, at least MathSciNet is behind a paywall unless you have a good IP address (so I would not say it is "the obvious thing to do"). Google Scholar is entirely free though.
– user138661
Apr 21, 2019 at 19:39
• ZBMath is free and knows 12 citations to the foundations book (but only shows three works without subscription: zbmath.org/?q=rf%3A0969.14013)
– Dirk
Apr 21, 2019 at 20:06

From MathSciNet:

MR3905130 Lan, Guitang ; Sheng, Mao ; Yang, Yanhong ; Zuo, Kang . Uniformization of p-adic curves via Higgs–de Rham flows. J. Reine Angew. Math. 747 (2019), 63--108.

R3700485 Wakabayashi, Yasuhiro . Duality for dormant opers. J. Math. Sci. Univ. Tokyo 24 (2017), no. 3, 271--320.

MR3658210 Joshi, Kirti . The degree of the dormant operatic locus. Int. Math. Res. Not. IMRN 2017, no. 9, 2599--2613.

MR3615581 Zhao, Yifei . Maximally Frobenius-destabilized vector bundles over smooth algebraic curves. Internat. J. Math. 28 (2017), no. 2, 1750003, 26 pp.

MR3417530 Hoshi, Yuichiro . Nilpotent admissible indigenous bundles via Cartier operators in characteristic three. Kodai Math. J. 38 (2015), no. 3, 690--731.

MR3318144 Joshi, Kirti ; Pauly, Christian . Hitchin-Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves. Adv. Math. 274 (2015), 39--75.

MR3296806 Gabber, Ofer ; Gille, Philippe ; Moret-Bailly, Laurent . Fibrés principaux sur les corps valués henséliens. (French) [[Principal bundles over Henselian valued fields]] Algebr. Geom. 1 (2014), no. 5, 573--612.

MR3262443 Wakabayashi, Yasuhiro . An explicit formula for the generic number of dormant indigenous bundles. Publ. Res. Inst. Math. Sci. 50 (2014), no. 3, 383--409.

MR3103899 Bouw, Irene I. ; Zapponi, Leonardo . Existence of covers with fixed ramification in positive characteristic. Int. J. Number Theory 9 (2013), no. 6, 1475--1489.

MR2680418 Bouw, Irene I. ; Möller, Martin . Teichmüller curves, triangle groups, and Lyapunov exponents. Ann. of Math. (2) 172 (2010), no. 1, 139--185.

MR2566970 Ducrohet, Laurent . The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 4, 1641--1669.

MR2518167 Osserman, Brian . Logarithmic connections with vanishing p-curvature. J. Pure Appl. Algebra 213 (2009), no. 9, 1651--1664.

MR2384903 Joshi, Kirti . Two remarks on subvarieties of moduli spaces. Internat. J. Math. 19 (2008), no. 2, 237--243.

MR2365412 Lange, Herbert ; Pauly, Christian . On Frobenius-destabilized rank-2 vector bundles over curves. Comment. Math. Helv. 83 (2008), no. 1, 179--209.

MR2317114 Osserman, Brian . Mochizuki's crys-stable bundles: a lexicon and applications. Publ. Res. Inst. Math. Sci. 43 (2007), no. 1, 95--119.

MR2285248 Benedetto, Robert L. Wandering domains in non-Archimedean polynomial dynamics. Bull. London Math. Soc. 38 (2006), no. 6, 937--950.

MR2266885 Conrad, Brian . Relative ampleness in rigid geometry. Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1049--1126.

MR2255181 Osserman, Brian . The generalized Verschiebung map for curves of genus 2. Math. Ann. 336 (2006), no. 4, 963--986.

MR2231194 Joshi, Kirti ; Ramanan, S. ; Xia, Eugene Z. ; Yu, Jiu-Kang . On vector bundles destabilized by Frobenius pull-back. Compos. Math. 142 (2006), no. 3, 616--630.

MR2223683 Liu, Fu ; Osserman, Brian . Mochizuki's indigenous bundles and Ehrhart polynomials. J. Algebraic Combin. 23 (2006), no. 2, 125--136.

MR2219211 Bouw, Irene I. ; Wewers, Stefan . Indigenous bundles with nilpotent p-curvature. Int. Math. Res. Not. 2006, Art. ID 89254, 37 pp.

MR2118045 Mochizuki, Shinichi . Categories of log schemes with Archimedean structures. J. Math. Kyoto Univ. 44 (2004), no. 4, 891--909.

MR2095769 Moriwaki, Atsushi . Diophantine geometry viewed from Arakelov geometry [translation of Sūgaku 54 (2002), no. 2, 113–129; MR1911908]. Sugaku Expositions. Sugaku Expositions 17 (2004), no. 2, 219--234.

MR1859246 Ogus, Arthur . Elliptic crystals and modular motives. Adv. Math. 162 (2001), no. 2, 173--216.

MR1834911 Nakamura, Hiroaki ; Tamagawa, Akio ; Mochizuki, Shinichi . The Grothendieck conjecture on the fundamental groups of algebraic curves [translation of Sūgaku 50 (1998), no. 2, 113–129; MR1648427 (2000e:14038)]. Sugaku Expositions. Sugaku Expositions 14 (2001), no. 1, 31--53.

MR1812812 Edixhoven, S. J. ; Moonen, B. J. J. ; Oort, F. Open problems in algebraic geometry. Bull. Sci. Math. 125 (2001), no. 1, 1--22.