21
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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). $\endgroup$ – Misha Apr 20 at 13:44
  • 1
    $\begingroup$ 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. $\endgroup$ – Dirk Apr 20 at 15:57
  • 1
    $\begingroup$ @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. $\endgroup$ – schematic_boi Apr 21 at 19:39
  • 1
    $\begingroup$ ZBMath is free and knows 12 citations to the foundations book (but only shows three works without subscription: zbmath.org/?q=rf%3A0969.14013) $\endgroup$ – Dirk Apr 21 at 20:06
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.