Hodge-Arakelov. Grothendieck. Neukirch- ([3], [4]) Grothendieck. survey Hodge-Arakelov. étale site topos, étale

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1 Hodge-Arakelov Grothendieck Diophantus Grothendieck Neukirch- ([3], [4]) Grothendieck survey Hodge-Arakelov Grothendieck Diophantus Hartshorne [1] Milne [2] étale site topos, étale Hodge-Arakelov basepoint, core, commensurator universe, [1] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer (1997). [2] J. S. Milne, Étale Cohomology, Princeton Mathematical Series 33, Princeton University Press (1980). [3] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio Math. 109 (1997), pp [4] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138

2 base parametrize Kodaira-Spencer base fiber Kodaira-Spencer Diophantus base Kodaira-Spencer Diophantus Kodaira-Spencer Diophantus [1] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138 [2] S. Mochizuki, The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories, RIMS Preprint Nos. 1255, 1256 (October 1999), available as.ps file at: kenkyubu/paper/all.html

3 1. Hodge-Arakelov [1] M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves, Journ. Pure Appl. Algebra 159 (2001), pp [2] J. W. S. Cassels and A. Fröhlich (editors), Algebraic number theory, Proceedings of the instructional conference held at the University of Sussex, Brighton, September 1-17, 1965, Academic Press (1986). [3] P. Deligne and D. Mumford, The Irreducibility of the Moduli Space of Curves of Given Genus, IHES Publ. Math. 36 (1969), pp [4] 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). [5] M. Matsumoto, Galois representations on profinite braid groups on curves, J. Reine Angew. Math. 474 (1996), pp [6] S. Mochizuki, The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories, RIMS Preprint Nos. 1255, 1256 (October 1999). [7] S. Mochizuki, The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve, RIMS Preprint No (July 2000). [8] S. Mochizuki, The Hodge-Arakelov Theory of Elliptic Curves in Positive Characteristic, RIMS Preprint No (October 2000). [9] S. Mochizuki, Correspondences on Hyperbolic Curves, Journ. Pure Appl. Algebra 131 (1998), pp [10] S. Mochizuki, A Version of the Grothendieck Conjecture for p-adic Local Fields, The International Journal of Math. 8 (1997), pp [11] S. Mochizuki, The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves over Number Fields, J. Math. Sci., Univ. Tokyo 3 (1996), pp [12] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138 [13] H. Nakamura, A. Tamagawa, and S. Mochizuki, The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves, Sugaku Expositions 14 (2001), pp [14] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag (2000). [15] J.-P. Serre, Local Class Field Theory in Algebraic Number Theory, ed. J.W.S. Cassels and A. Fröhlich, Academic Press (1967). 1

4 [16] J.-P. Serre, Trees, Springer-Verlag (1980). [17] J.-P. Serre (with the collaboration of Willem Kuyk and John Labute), Abelian l-adic Representations and Elliptic Curves, Addison-Wesley Publishing Company (1989). [18] Revêtement étales et groupe fondamental, Séminaire de Géometrie Algébrique du Bois Marie (SGA1), dirigé par A. Grothendieck, augmenté de deux exposés de M. Raynaud, Lecture Notes in Mathematics 224, Springer-Verlag (1971). [19] A. Tamagawa, The Grothendieck Conjecture for Affine Curves, Compositio Math. 109 (1997), pp [Hh] [Mn] [Hh] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer-Verlag (1977). [Mn] J. S. Milne, Étale Cohomology, Princeton Mathematical Series 33, Princeton University Press (1980). 5. Grothendieck Grothendieck Neukirch- ( [12], [19] ) Grothendieck survey Hodge-Arakelov Grothendieck Diophantus I Grothendieck anabelian II Hodge-Arakelov III basepoint, core, commensurator 1. anabelioid 2. core 3. 2

5 4. 5. global multiplicative subspace IV universe, orbicurve