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Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

Published online by Cambridge University Press:  11 January 2016

Yuichiro Hoshi
Yuichiro Hoshi*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japanyuichiro@kurims.kyoto-u.ac.jp
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Abstract

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Let l be a prime number. In this paper, we prove that the isomorphism class of an l-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-l outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

Keywords

14H3014H10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 203 , September 2011 , pp. 47 - 100
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

References

[1] Anderson, M. P., Exactness properties of profinite completion functors, Topology 13 (1974), 229–239.CrossRefGoogle Scholar
[2] Anderson, G. and Ihara, Y., Pro-l branched coverings of P1 and higher circular l-units, Ann. of Math. (2) 128 (1988), 271–293.CrossRefGoogle Scholar
[3] Asada, M., The faithfulness of the monodromy representations associated with certain families of algebraic curves, J. Pure Appl. Algebra 159 (2001), 123–147.CrossRefGoogle Scholar
[4] Chai, C. L. and Oort, F., A note on the existence of absolutely simple Jacobians, J. Pure Appl. Algebra 155 (2001), 115–120.CrossRefGoogle Scholar
[5] Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75–109.CrossRefGoogle Scholar
[6] Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique, III: Études cohomologique des faisceaux cohérents, I, Publ. Math. Inst. Hautes Études Sci. 11 (1961).Google Scholar
[7] Grothendieck, A. and Dieudonné, J., Groupes de monodromie en géométrie algébrique, I, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 7 I), Lecture Notes in Math. 288, Springer, Berlin, 1972.Google Scholar
[8] Hoshi, Y. and Mochizuki, S., On the combinatorial anabelian geometry of nodally nondegenerate outer representations, preprint, to appear in Hiroshima Math. J.Google Scholar
[9] Ihara, Y. and Nakamura, H., On deformation of maximally degenerate stable marked curves and Oda’s problem, J. Reine Angew. Math. 487 (1997), 125–151.Google Scholar
[10] Knudsen, F., The projectivity of the moduli space of stable curves, II: The stacks Mg,n , Math. Scand. 52 (1983), 161–199.CrossRefGoogle Scholar
[11] Matsumoto, M. and Tamagawa, A., Mapping-class-group action versus Galois action on profinite fundamental groups, Amer. J. Math. 122 (2000), 1017–1026.CrossRefGoogle Scholar
[12] Mochizuki, S., Correspondences on hyperbolic curves, J. Pure Appl. Algebra 131 (1998), 227–244.CrossRefGoogle Scholar
[13] Mochizuki, S., The local pro-p anabelian geometry of curves, Invent. Math. 138 (1999), 319–423.CrossRefGoogle Scholar
[14] Mochizuki, S., “Topics surrounding the anabelian geometry of hyperbolic curves” in Galois Groups and Fundamental Groups, Math. Sci. Res. Inst. Publ. 41, Cambridge University Press, Cambridge, 2003, 119–165.Google Scholar
[15] Mochizuki, S., “The absolute anabelian geometry of hyperbolic curves” in Galois Theory and Modular Forms, Dev. Math. 11, Kluwer, Boston, 2004, 77–122.Google Scholar
[16] Mochizuki, S. and Tamagawa, A., The algebraic and anabelian geometry of configuration spaces, Hokkaido Math. J. 37 (2008), 75–131.CrossRefGoogle Scholar
[17] Nakamura, H., Galois rigidity of pure sphere braid groups and profinite calculus, J. Math. Sci. Univ. Tokyo 1 (1994), 71–136.Google Scholar
[18] Nakamura, H., Takao, N., and Ueno, R., Some stability properties of Teichmüller modular function fields with pro-l weight structures, Math. Ann. 302 (1995), 197–213.CrossRefGoogle Scholar
[19] Neukirch, J., Algebraic Number Theory, Grundlehren Math. Wiss. 322, Springer, Berlin, 1999.Google Scholar
[20] Oda, T., “Etale homotopy type of the moduli spaces of algebraic curves” in Geometric Galois Actions, Vol. 1, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, Cambridge, 1997, 85–95.Google Scholar
[21] Poonen, B., Varieties without extra automorphisms, II: Hyperelliptic curves, Math.Res. Lett. 7 (2000), 77–82.CrossRefGoogle Scholar
[22] Ribes, L. and Zalesskii, P., Profinite Groups, Ergeb. Math. Grenzgeb. (3) 40, Springer, Berlin, 2000.Google Scholar
[23] Serre, J. P. and Tate, J., Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517.CrossRefGoogle Scholar
[24] Takao, N., Braid monodromies on proper curves and pro-l Galois representations, preprint, to appear in J. Inst. Math. Jussieu.Google Scholar
[25] Tamagawa, A., The Grothendieck conjecture for affine curves, Compos. Math. 109 (1997), 135–194.CrossRefGoogle Scholar
[26] Tamagawa, A., Resolution of nonsingularities of families of curves, Publ. Res. Inst. Math. Sci. 40 (2004), 1291–1336.Google Scholar