Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves

[Andr´e] Y. Andr´e, On a geometric description of Gal(Qp /Qp ) and a p-adic

d , Duke Math. J. 119 (2003), pp. 1–39.

avatar of GT

[Brk] V. G. Berkovich, Spectral theory and analytic geometry over nonArchimedean fields, Mathematical Surveys and Monographs, 33, American

Mathematical Society (1990).

[BLR] S. Bosch, W. L¨

utkebohmert, and M. Raynaud, N´eron Models, Ergebnisse

der Mathematik und ihrer Grenzgebiete 21, Springer-Verlag (1990).

[DM] P. Deligne and D. Mumford, The irreducibility of the space of curves of

given genus, IHES Publ. Math. 36 (1969), pp. 75–109.

103

[EP] A. Engler and A. Prestel, Valued Fields, Springer Monographs in Mathematics, Springer-Verlag (2005).

[FC] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Ergebnisse

der Mathematik und ihrer Grenzgebiete 22, Springer-Verlag (1990).

´ ements de g´eometrie alg´ebrique. IV: Etude

[EGAIV2 ] A. Grothendieck, El´

locale

des sch´emas et des morphismes de sch´emas, IHES Publ. Math. 24 (1965).

´ ements de g´eometrie alg´ebrique. IV: Etude

[EGAIV3 ] A. Grothendieck, El´

locale

des sch´emas et des morphismes de sch´emas, IHES Publ. Math. 28 (1966).

[Hgsh] K. Higashiyama, The semi-absolute anabelian geometry of geometrically

pro-p arithmetic fundamental groups of associated low-dimensional configuration spaces, Publ. Res. Inst. Math. Sci. 58 (2022), pp. 579–634.

[BSC] Y. Hoshi, Conditional results on the birational section conjecture over

small number fields, Automorphic forms and Galois representations, Vol. 2,

London Math. Soc. Lecture Note Ser. 415, Cambridge Univ. Press (2014),

pp. 187–230.

[NodNon] Y. Hoshi and S. Mochizuki, On the combinatorial anabelian geometry

of nodally nondegenerate outer representations, Hiroshima Math. J. 41

(2011), pp. 275–342.

[CbTpI] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: Inertia groups and profinite Dehn

twists, Galois-Teichm¨

uller Theory and Arithmetic Geometry, Adv. Stud.

Pure Math. 63, Math. Soc. Japan (2012), pp. 659–811.

[CbTpII] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial anabelian geometry of hyperbolic curves II: Tripods and combinatorial cuspidalization, Lecture Notes in Mathematics 2299, Springer-Verlag (2022).

[CbTpIII] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial

anabelian geometry of hyperbolic curves III: Tripods and tempered fundamental groups, RIMS Preprint 1763 (November 2012).

[CbTpIV] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial

anabelian geometry of hyperbolic curves IV: Discreteness and sections, to

appear in Nagoya Math. J.

[Knud] F. F. Knudsen, The projectivity of the moduli space of stable curves,

II: The stacks Mg,n , Math. Scand. 52 (1983), pp. 161–199.

[HMM] Y. Hoshi, A. Minamide, and S. Mochizuki, Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups,

Kodai Math. J. 45 (2022), pp. 295–348.

104

[HMT] Y. Hoshi, S. Mochizuki, and S. Tsujimura, Combinatorial construction

of the absolute Galois group of the field of rational numbers, RIMS Preprint

1935 (December 2020).

[Lpg1] E. Lepage, Resolution of non-singularities for Mumford curves, Publ.

Res. Inst. Math. Sci. 49 (2013), pp. 861–891.

[Lpg2] E. Lepage, Resolution of non-singularities and the absolute anabelian

conjecture, arXiv:2306.07058 [math.AG].

[Milne] J. S. Milne, Jacobian varieties in Arithmetic Geometry, ed. by G. Cornell

and J.H. Silverman, Springer-Verlag (1986), pp. 167–212.

[MiSaTs] A. Minamide, K. Sawada, and S. Tsujimura, On generalizations of

anabelian group-theoretic properties, RIMS Preprint 1965 (August 2022).

[MiTs1] A. Minamide and S. Tsujimura, Anabelian group-theoretic properties

of the absolute Galois groups of discrete valuation fields, J. Number Theory

239 (2022), pp. 298–334.

[MiTs2] A. Minamide and S. Tsujimura, Anabelian geometry for Henselian discrete valuation fields with quasi-finite residues, RIMS Preprint 1973 (June

2023).

[Hur] S. Mochizuki, The geometry of the compactification of the Hurwitz

scheme, Publ. Res. Inst. Math. Sci. 31 (1995), pp. 355–441.

[PrfGC] S. Mochizuki, The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Univ. Tokyo 3 (1996), pp.

571–627.

[LocAn] S. Mochizuki, The local pro-p anabelian geometry of curves, Invent.

Math. 138 (1999), pp. 319–423.

[ExtFam] S. Mochizuki, Extending families of curves over log regular schemes,

J. reine angew. Math. 511 (1999), pp. 43–71.

[CanLift] S. Mochizuki, The absolute anabelian geometry of canonical curves,

Kazuya Kato’s fiftieth birthday, Doc. Math. 2003, Extra Vol., pp. 609–640.

[AnabTop] S. Mochizuki, Topics surrounding the anabelian geometry of hyperbolic curves, Galois groups and fundamental groups, Math. Sci. Res. Inst.

Publ. 41, Cambridge Univ. Press. (2003), pp. 119–165.

[NCBel] S. Mochizuki, Noncritical Belyi maps, Math. J. Okayama Univ. 46

(2004), pp. 105–113.

[AbsAnab] S. Mochizuki, The absolute anabelian geometry of hyperbolic curves,

Galois Theory and Modular Forms, Kluwer Academic Publishers (2004),

pp. 77–122.

105

[SemiAn] S. Mochizuki, Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci.

42 (2006), pp. 221–322.

[CmbGC] S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math. J. 59 (2007), pp. 455–479.

[CmbCsp] S. Mochizuki, On the combinatorial cuspidalization of hyperbolic

curves, Osaka J. Math. 47 (2010), pp. 651–715.

[AbsTopI] S. Mochizuki, Topics in absolute anabelian geometry I: Generalities,

J. Math. Sci. Univ. Tokyo 19 (2012), pp. 139–242.

[AbsTopII] S. Mochizuki, Topics in absolute anabelian geometry II: Decomposition groups and endomorphisms, J. Math. Sci. Univ. Tokyo 20 (2013),

pp. 171–269.

[AbsTopIII] S. Mochizuki, Topics in absolute anabelian geometry III: Global

reconstruction algorithms, J. Math. Sci. Univ. Tokyo 22 (2015), pp. 939–

1156.

[MT] S. Mochizuki and A. Tamagawa, The algebraic and anabelian geometry

of configuration spaces, Hokkaido Math. J. 37 (2008), pp. 75–131.

[Neu] J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen

Wissenschaften 322, Springer-Verlag (1999).

[NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields,

Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag

(2000).

[Ray1] M. Raynaud, Sections des fibr´es vectoriels sur une courbe, Bull. Soc.

Math. France, 110 (1982), pp. 103–125.

[Ray2] M. Raynaud, p-groupes et r´eduction semi-stable des courbes, The

Grothendieck Festschrift, Vol. III, Progr. Math., 88, Birkh¨auser (1990),

pp. 179–197.

[Tama1] A. Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), pp. 135–194.

[Tama2] A. Tamagawa, Resolution of nonsingularities of families of curves, Publ.

Res. Inst. Math. Sci. 40 (2004), pp. 1291–1336.

[Tate] J. Tate, p-divisible groups, Proceedings of a Conference on Local Fields,

Driebergen, Springer-Verlag (1967), pp. 158–183.

[Tsjm] S. Tsujimura, Combinatorial Belyi cuspidalization and arithmetic subquotients of the Grothendieck-Teichm¨

uller group, Publ. Res. Inst. Math.

Sci. 56 (2020), pp. 779–829.

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Updated versions of [HMT], [CbTpIII], [CbTpIV] may be found at the following

URL:

http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html

(Shinichi Mochizuki) Research Institute for Mathematical Sciences, Kyoto

University, Kyoto 606-8502, Japan

Email address: motizuki@kurims.kyoto-u.ac.jp

(Shota Tsujimura) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Email address: stsuji@kurims.kyoto-u.ac.jp

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