リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

リケラボ 全国の大学リポジトリにある学位論文・教授論文を一括検索するならリケラボ論文検索大学・研究所にある論文を検索できる

大学・研究所にある論文を検索できる 「Grothendieck Conjecture for Hyperbolic Curves over Finitely Generated Fields of Positive Characteristic via Compatibility of Cyclotomes」の論文概要。リケラボ論文検索は、全国の大学リポジトリにある学位論文・教授論文を一括検索できる論文検索サービスです。

コピーが完了しました

URLをコピーしました

論文の公開元へ論文の公開元へ
書き出し

Grothendieck Conjecture for Hyperbolic Curves over Finitely Generated Fields of Positive Characteristic via Compatibility of Cyclotomes

TSUJIMURA, Shota 京都大学

2023.07

概要

Let p be a prime number. In the present paper, from the viewpoint of the compatibility/rigidity of group-theoretic cyclotomes, we revisit the anabelian Grothendieck Conjecture for hyperbolic curves over finitely generated fields of characteristic p established by A. Tamagawa, J. Stix, and S. Mochizuki. Especially, we give an alternative proof of the Grothendieck Conjecture for nonisotrivial hyperbolic curves over finitely generated fields of characteristic p obtained by J. Stix. In fact, by applying relatively recent results in anabelian geometry for hyperbolic curves over finite fields developed by M. Saïdi and A. Tamagawa, we discuss the J. Stix's result in a certain generalized situation, i.e., the geometrically pro-Σ setting, where Σ denotes the complement of a finite set of prime numbers that contains p in the set of all prime numbers. Moreover, by combining with a theorem in birational anabelian geometry obtained by F. Pop, we prove an absolute version of the geometrically pro-Σ Grothendieck Conjecture for nonisotrivial hyperbolic curves over the perfections of finitely generated fields of characteristic p. On the other hand, in the present paper, we also establish certain isotriviality criteria for hyperbolic curves with respect to both l-adic Galois representations and pro-l outer Galois representations, where l is a prime number ≠ p. These isotriviality criteria may be applied to give an alternative proof of the J. Stix's result.

参考文献

[1] B. Conrad, Chow’s K/k-image and K/k-trace, and the Lang-N´eron theorem, L’Enseignement

Math´ematique 52 (2006), pp. 37–108.

[2] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, IHES Publ.

Math. 36 (1969), pp. 75–109.

[3] J. Fogarty, F. Kirwan, and D. Mumford, Geometric Invariant Theory, Ergebnisse der Mathematik

und ihrer Grenzgebiete 34, Springer-Verlag (1994).

[4] M. Fried and M. Jarden, Field Arithmetic (Second edition), Ergebnisse der Mathematik und ihrer

Grenzgebiete 3. Folge, A Series of Modern Surveys in Mathematics 11, Springer-Verlag (2005).

[5] A. Grothendieck, Letter to G. Faltings (June 1983) in Lochak, L. Schneps, Geometric Galois Actions; 1. Around Grothendieck’s Esquisse d’un Programme, London Math. Soc. Lect. Note Ser. 242,

Cambridge Univ. Press (1997).

[6] A. Grothendieck and J. Murre, The tame fundamental group of a formal neighbourhood of a divisor

with normal crossings on a scheme, Lecture Notes in Math. 208 (1971), Springer-Verlag.

[7] A. Grothendieck and M. Raynaud, Revˆetements ´etales et groupe fondamental (SGA1), Lecture Notes

in Math. 224 (1971), Springer-Verlag.

[8] F. J. Grunewald and D. Segal, On congruence topologies in number fields, J. Reine Angew. Math.

311/312 (1979), pp. 389–396.

[9] Y. Hoshi and S. Tsujimura, On the injectivity of the homomorphisms from the automorphism groups

of fields to the outer automorphism groups of the absolute Galois groups, Res. Number Theory 9,

Paper No. 44 (2023).

[10] S. Lang and A. N´eron, Rational points of abelian varieties over function field, Amer. J. Math. 81

(1959), pp. 95–118.

[11] H. Martens, A new proof of Torelli’s theorem, Ann. Math. 63 (1963), pp. 107–111.

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

Springer-Verlag (1986), pp. 167–212.

[13] A. Minamide, Indecomposability of various profinite groups arising from hyperbolic curves, Okayama

Math. J. 60 (2018), pp. 175–208.

[14] S. Mochizuki, The profinite Grothendieck conjecture for closed hyperbolic curves over number fields,

J. Math. Sci. Univ. Tokyo 3 (1996), pp. 571–627.

[15] S. Mochizuki, The local pro-p anabelian geometry of curves, Invent. Math. 138 (1999), pp. 319–423.

[16] S. Mochizuki, Topics surrounding the anabelian geometry of hyperbolic curves, Galois groups and

fundamental groups, Math. Sci. Res. Inst. Publ. 41, Cambridge Univ. Press, Cambridge, (2003), pp.

119–165.

[17] S. Mochizuki, The absolute anabelian geometry of hyperbolic curves, Galois theory and modular

forms, Kluwer Academic Publishers (2004), pp. 77–122.

[18] S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math. J. 59 (2007),

pp. 455–479.

29

[19] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ.

47 (2007), pp. 451–539.

[20] S. Mochizuki, Topics in absolute anabelian geometry I: Generalities, J. Math. Sci. Univ. Tokyo 19

(2012), pp. 139–242.

[21] S. Mochizuki and A. Tamagawa, The algebraic and anabelian geometry of configuration spaces,

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

[22] D. Mumford, Abelian varieties, Oxford Univ. Press (1974).

[23] F. Pop, Alterations and birational anabelian geometry, in H. Hauser, J. Lipman, F. Oort, A. Quir´os,

Resolution of Singularities, Progress in Math. 181, pp. 519–533, Birkh¨auser-Verlag, Basel (2000).

[24] P. Samuel, Lectures on old and new results on algebraic curves, notes by S. Anantharaman, Tata

Institute of Fundamental Research Lectures on Mathematics. 36 (1966).

[25] M. Sa¨ıdi and A. Tamagawa, A prime-to-p version of Grothendieck’s anabelian conjecture for hyperbolic curves over finite fields of characteristic p > 0, Publ. Res. Inst. Math. Sci. 45 (2009), pp.

135–186.

[26] M. Sa¨ıdi and A. Tamagawa, A refined version of Grothendieck’s anabelian conjecture for hyperbolic

curves over finite fields, J. Algebraic Geom. 27 (2018), pp. 383–448.

[27] J. Stix, Affine anabelian curves in positive characteristic, Compositio Math. 134 (2002), pp. 75–85.

[28] J. Stix, Projective anabelian curves in positive characteristic and descent theory for log-´etale covers,

Dissertation, Bonn, Bonner Mathematische Schriften 354 (2002).

[29] A. Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), pp.

135–194.

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

Japan

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

30

...

参考文献をもっと見る

全国の大学の
卒論・修論・学位論文

一発検索!

この論文の関連論文を見る