Hostname: page-component-f554764f5-44mx8 Total loading time: 0 Render date: 2025-04-22T07:24:23.341Z Has data issue: false hasContentIssue false
  • English
  • Français

ON ODA’S PROBLEM AND SPECIAL LOCI

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Curves

Published online by Cambridge University Press:  12 November 2024

Benjamin Collas Open the ORCID record for Benjamin Collas [Opens in a new window]  and
Séverin Philip
Benjamin Collas*
Affiliation:
Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502, Japan ([email protected])
Séverin Philip
Affiliation:
Research Institute for Mathematical Sciences Kyoto University, Kyoto 606-8502, Japan ([email protected])
*
[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Oda’s problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda’s prediction.

Keywords

Oda’s problemmoduli space of curvesspecial lociuniversal monodromy action

MSC classification

Primary: 14H30: Coverings, fundamental group 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields
Secondary: 14H10: Families, moduli (algebraic) 14H25: Arithmetic ground fields 14G32: Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 24 , Issue 2 , March 2025 , pp. 503 - 539
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Supported by JSPS KAKENHI Grant Number 22F22015 (S. Philip) and by the Research Institute for Mathematical Sciences, Kyoto University. This work is part of the “Arithmetic and Homotopic Galois Theory” project, supported by the CNRS France-Japan AHGT International Research Network between the RIMS Kyoto University, the LPP of Lille University, and the DMA of ENS PSL.

References

Anderson, M. P., Exactness properties of profinite completion functors. Topology 13 (1974), 229–239. ISSN 0040-9383. doi: https://doi.org/10.1016/0040-9383(74)90016-0.CrossRefGoogle Scholar
Bertin, J. and Romagny, M., Hurwitz stacks. Mém. Soc. Math. Fr., Nouv. Sér. 125–126 (2011), 219. doi: 10.24033/msmf.437.Google Scholar
Collas, B., Action of a Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups of genus one. Int. J. Number Theory 8(3) (2012), 763–787. doi: 10.1142/S1793042112500431.CrossRefGoogle Scholar
Collas, B. and Maugeais, S., Composantes irréductibles de lieux spéciaux d’espaces de modules de courbes, action galoisienne en genre quelconque. Ann. Inst. Fourier 65(1) (2015), 245–276. doi: 10.5802/aif.2930.CrossRefGoogle Scholar
Collas, B. and Maugeais, S., On Galois action on the inertia stack of moduli spaces of curves. Publ. Res. Inst. Math. Sci. 59(4) (2023), 731–758. https://doi.org/10.4171/prims/59-4-2.CrossRefGoogle Scholar
Ekedahl, T., Boundary behaviour of Hurwitz schemes. In The Moduli Space of Curves. Proceedings of the Conference Held on Texel Island, Netherlands During the Last Week of April 1994. (Basel: Birkhäuser, 1995), 173–198. https://doi.org/10.1007/978-1-4612-4264-2_7.CrossRefGoogle Scholar
González-Diez, G. and Hidalgo, R. A., Conformal versus topological conjugacy of automorphisms on compact Riemann surfaces. Bull. Lond. Math. Soc. 29(3) (1997), 280–284. doi: 10.1112/S0024609396002640.CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. (4) (1960), 228.CrossRefGoogle Scholar
Grothendieck, A., Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. Inst. Hautes Études Sci. Publ. Math. (17) (1963), 137–223.Google Scholar
Grothendieck, A. and Murre, J. P., The Tame Fundamental Groups of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, vol. 208. Lect. Notes Math. New York: Springer. https://doi.org/10.1007/BFb0069608.Google Scholar
Hoshi, Y., Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero. Nagoya Math. J. 203 (2011), 47–100. doi: 10.1215/00277630-1331863.CrossRefGoogle Scholar
Hoshi, Y., On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves: genus zero case. Tohoku Math. J. (2) 65(2) (2013), 231–242. doi: 10.2748/tmj/1372182723.CrossRefGoogle Scholar
Hoshi, Y. and Mochizuki, S., On the combinatorial anabelian geometry of nodally nondegenerate outer representations. Hiroshima Math. J. 41(3) (2011), 275–342. doi: 10.32917/hmj/1323700038.CrossRefGoogle Scholar
Ihara, Y., Profinite braid groups, Galois representations and complex multiplications. Ann. of Math. (2) 123(1) (1986), 43–106. doi: 10.2307/1971352.CrossRefGoogle Scholar
Ihara, Y., The Galois representation arising from $\mathbb {P}^1$ − {0, 1, ∞} and Tate twists of even degree. In Galois Groups Over  $\mathbb {Q}$ (Berkeley, CA, 1987), vol. 16. (Math. Sci. Res. Inst. Publ. New York: Springer, 1989), 299–313. doi: 10.1007/978-1-4613-9649-9_4.CrossRefGoogle Scholar
Ihara, Y., On the embedding of Gal(Q/Q) intô GT. In The Grothendieck Theory of Dessins D’enfants (Luminy, 1993), vol. 200. London Math. Soc. Lecture Note Ser. (Cambridge: Cambridge Univ. Press, 1994), 289–321. With an appendix: the action of the absolute Galois group on the moduli space of spheres with four marked points by Michel Emsalem and Pierre Lochak.Google Scholar
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. https://doi.org/10.1515/crll.1997.487.125.Google Scholar
Knudsen, F. F., The projectivity of the moduli space of stable curves. II: The stacks Mg,n. Math. Scand. 52 (1983), 161–199. http://eudml.org/doc/166839.CrossRefGoogle Scholar
Magaard, K. (2002), The locus of curves with prescribed automorphism group. Sūrikaisekikenkyūsho Kōkyūroku Shaska, T., Shpectorov, S. and Völklein, H., (1267), 112–141. Communications in arithmetic fundamental groups (Kyoto, 1999/2001).Google Scholar
Matsumoto, M., Galois representations on profinite braid groups on curves. J. Reine Angew. Math. 474 (1996), 160–219. https://doi.org/10.1515/crll.1996.474.169.Google Scholar
Mochizuki, S. and Tamagawa, A., The algebraic and anabelian geometry of configuration spaces. Hokkaido Math. J. 37(1) (2008), 75–131. doi: 10.14492/hokmj/1253539588.CrossRefGoogle Scholar
Morita, S., The extension of Johnson’s homomorphism from the Torelli group to the mapping class group. Invent. Math. 111(1) (1993), 197–224. doi: 10.1007/BF01231286.CrossRefGoogle Scholar
Nakamura, H., Coupling of universal monodromy representations of Galois- Teichmüller modular groups. Math. Ann. 304(1) (1996), 99–119. doi: 10.1007/BF01446287.CrossRefGoogle Scholar
Nakamura, H., Tangential base points and Eisenstein power series. In Aspects of Galois Theory. Papers from the Conference on Galois theory, Gainesville, FL, USA, October 14–18, 1996. (Cambridge: Cambridge University Press, 1999), 202–217.Google Scholar
Nakamura, H., Takao, N. and Ueno, R., Some stability properties of Teichmüller modular function fields with pro-l weight structures. Math. Ann. 302(2) (1995), 197–213. doi: 10.1007/BF01444493.CrossRefGoogle Scholar
Noohi, B., Fundamental groups of algebraic stacks. J. Inst. Math. Jussieu 3(1) (2004), 69–103. doi: 10.1017/S1474748004000039.CrossRefGoogle Scholar
Oda, T., The universal monodromy representations on the pro-nilpotent fundamental groups of algebraic curves. Mathematische Arbeitstagung (Neue Serie), Max-Planck- Institut preprint MPI/93-57, 9-15, 6.Google Scholar
Rasmussen, C. and Tamagawa, A., Arithmetic of abelian varieties with constrained torsion. Trans. Amer. Math. Soc. 369(4) (2017), 2395–2424. doi: 10.1090/tran/6790.CrossRefGoogle Scholar
Romagny, M., Connected and irreducible components in families. Manuscr. Math. 136(1–2) (2011), 1–32. doi: 10.1007/s00229-010-0424-7.CrossRefGoogle Scholar
Seyama, A., On the curves of genus g with automorphisms of prime order 2g + 1. Tsukuba J. Math. 6 (1982), 67–77. https://www.jstor.org/stable/43685303.CrossRefGoogle Scholar
Takao, N., Braid monodromies on proper curves and pro-ℓ Galois representations. J. Inst. Math. Jussieu 11(1) (2012), 161–181. doi: 10.1017/S1474748010000241.CrossRefGoogle Scholar
Takao, N., Some remarks on field towers arising from pro-nilpotent universal monodromy representations. RIMS Kôkyûroku Bessatsu B51 (2014), 55–70.Google Scholar
Tsunogai, H., Some new-type equations in the Grothendieck-Teichmüller group arising from geometry M0,5. In Primes and Knots. Proceedings of an AMS Special Session, Baltimore, MD, USA, January 15–16, 2003 and the 15th JAMI (Japan-US Mathematics Institute) Conference, Baltimore, MD, USA, March 7–16, 2003. Providence, (RI: American Mathematical Society, 2006), 263–284. https://doi.org/10.1090/conm/416.Google Scholar
Ueno, R., A Stability of the Galois Kernel Arising from Pro-ℓ Monodromy Representation. Master thesis, (RIMS, Kyoto University 1994).Google Scholar