Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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154 SHINICHI MOCHIZUKI to be a triple, consisting of the following data: (a) a D-Θ ±ell -Hodge theater † HT D-Θ±ell [cf. Definition 6.4, (iii)]; (b) a D-ΘNF-Hodge theater † HT D-ΘNF [cf. Definition 4.6, (iii)]; (c) the [necessarily unique!] gluing isomorphism between † HT D-Θ±ell and † HT D-ΘNF [cf. the discussion of Remark 6.12.2, (i), (ii)]. An illustration of the combinatorial structure of a D-Θ ±ell NF-Hodge theater is given in Fig. 6.5 above.
INTER-UNIVERSAL TEICHMÜLLER THEORY I 155 Bibliography [André] Y. André, On a Geometric Description of Gal(Q p /Q p )andap-adic Avatar of ĜT , Duke Math. J. 119 (2003), pp. 1-39. [Asada] M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves, J. Pure Appl. Algebra 159 (2001), pp. 123-147. [Falt] G. Faltings, Endlichkeitssätze für Abelschen Varietäten über Zahlkörpern, Invent. Math. 73 (1983), pp. 349-366. [FRS] B. Fine, G. Rosenberger, and M. Stille, Conjugacy pinched and cyclically pinched one-relator groups, Rev. Mat. Univ. Complut. Madrid 10 (1997), pp. 207-227. [Groth] 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). [NodNon] Y. Hoshi, S. Mochizuki, On the Combinatorial Anabelian Geometry of Nodally Nondegenerate Outer Representations, Hiroshima Math. J. 41 (2011), pp. 275- 342. [JP] K. Joshi and C. Pauly, Hitchin-Mochizuki morphism, opers, and Frobeniusdestabilized vector bundles over curves, preprint. [Kim] M. Kim, The motivic fundamental group of P 1 \{0, 1, ∞} and the theorem of Siegel, Invent. Math. 161 (2005), pp. 629-656. [Lang] S. Lang, Algebraic number theory, Addison-Wesley Publishing Co. (1970). [Lehto]O.Lehto,Univalent Functions and Teichmüller Spaces, Graduate Texts in Mathematics 109, Springer, 1987. [PrfGC] S. Mochizuki, The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves over Number Fields, J. Math. Sci. Univ. Tokyo 3 (1996), pp. 571-627. [pOrd] S. Mochizuki, A Theory of Ordinary p-adic Curves, Publ. Res. Inst. Math. Sci. 32 (1996), pp. 957-1151. [pTeich] S. Mochizuki, Foundations of p-adic Teichmüller Theory, AMS/IP Studies in Advanced Mathematics 11, American Mathematical Society/International Press (1999). [pGC] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138 (1999), pp. 319-423. [HASurI] S. Mochizuki, A Survey of the Hodge-Arakelov Theory of Elliptic Curves I, Arithmetic Fundamental Groups and Noncommutative Algebra, Proceedings of Symposia in Pure Mathematics 70, American Mathematical Society (2002), pp. 533-569. [HASurII] S. Mochizuki, A Survey of the Hodge-Arakelov Theory of Elliptic Curves II, Algebraic Geometry 2000, Azumino, Adv. Stud. Pure Math. 36, Math. Soc. Japan (2002), pp. 81-114.
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associated to † F ⊩ mod and ‡
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completely determined by † D. INT
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conjugation by which maps φ NF IN
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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