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An effective Arakelov-theoretic version of the hyperbolic isogeny theorem

Published online by Cambridge University Press:  14 January 2016

ARIYAN JAVANPEYKAR
ARIYAN JAVANPEYKAR*
Affiliation:
Institut für Mathematik, Johannes Gutenberg-Universität, Mainz, Germany. e-mail: ariyanjavan@gmail.com
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Abstract

For any integer e and hyperbolic curve X over $\overline{\mathbb Q}$, Mochizuki showed that there are only finitely many isomorphism classes of hyperbolic curves Y of Euler characteristic e with the same universal cover as X. We use Arakelov theory to prove an effective version of this finiteness statement.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 3 , May 2016 , pp. 463 - 476
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1]Abbes, A. and Ullmo, E.Auto-intersection du dualisant relatif des courbes modulaires X0(N). J. Reine Angew. Math. 484 (1997), 1–70.Google Scholar
[2]Alsina, M. and Bayer, P.Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series. Amer. Math. Soc., vol. 22. (Providence, RI, 2004).Google Scholar
[3]Behrend, K. and Noohi, B.Uniformisation of Deligne–Mumford curves. J. Reine Angew. Math. 599 (2006), 111–153.Google Scholar
[4]Belyi, G. V.Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43 (2) (1979), 267–276, 479.Google Scholar
[5]Bilu, Y. F. and Strambi, M.Quantitative Riemann existence theorem over a number field. Acta Arith. 145 (4) (2010), 319–339.CrossRefGoogle Scholar
[6]Bost, J.-B.Périodes et isogenies des variétés abéliennes sur les corps de nombres (daprès D. Masser et G. Wüstholz). Astérisque. (237), No. 795, 4, (1996), 115–161. Séminaire Bourbaki vol. 1994/95.Google Scholar
[7]de Jong, R.Arakelov invariants of Riemann surfaces. Doc. Math. 10 (electronic), (2005), 311–329.Google Scholar
[8]Deligne, P.Le groupe fondamental de la droite projective moins trois points. In: Galois Groups Over Q (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ. vol. 16. (Springer, New York, 1989), pp. 79–297.Google Scholar
[9]Edixhoven, B. and de Jong, R.Bounds for Arakelov invariants of modular curves. In: Computational Aspects of Modular Forms and Galois Representations. Ann. of Math. Stud. vol. 176. (Princeton University Press, Princeton, NJ, 2011), pp. 217–256.Google Scholar
[10]Faltings, G.Endlichkeitssätze für abelsche Varietäten ber Zahlkörpern. Invent. Math. 73 (3) (1983), 349–366.Google Scholar
[11]Faltings, G.Calculus on arithmetic surfaces. Ann. of Math. (2), 119 (2) (1984), 387–424.CrossRefGoogle Scholar
[12]Gaudron, É. and Rémond, G.Théorème des pèriodes et degrés minimaux d'isogénies. Comment. Math. Helv. 89 (2) (2014), 343–403.Google Scholar
[13]Javanpeykar, A.Polynomial bounds for Arakelov invariants of Belyi curves. Algebra Number Theory. 8 (1) (2014), 89–140. With an appendix by Peter Bruin.Google Scholar
[14]Jorgenson, J. and Kramer, J.Bounding the sup-norm of automorphic forms. Geom. Funct. Anal. 14 (6) (2004), 1267–1277.Google Scholar
[15]Jorgenson, J. and Kramer, J.Bounds on Faltings's delta function through covers. Ann. of Math. (2), 170 (1) (2009), 1–43.Google Scholar
[16]Margulis, G. A.Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. (3) vol. 17. (Springer Verlag, Berlin, 1991).Google Scholar
[17]Masser, D. and Wüstholz, G.Isogeny estimates for abelian varieties and finiteness theorems. Ann. of Math. (2), 137 (3) (1993), 459–472.Google Scholar
[18]Masser, D. and Wüstholz, G.Periods and minimal abelian subvarieties. Ann. of Math. (2), 137 (2) (1993), 407–458.Google Scholar
[19]Mayer, H.Self-intersection of the relative dualising sheaf on modular curves X1(N). J. Théor. Nombres Bordeaux 26 (1) (2014), 111–161.Google Scholar
[20]Merkl, F.An upper bound for Green functions on Riemann surfaces. In: Computational Aspects of Modular Forms and Galois Representations. Ann. of Math. Stud. vol. 176. (Princeton University Press, Princeton, NJ, 2011), 203–215.Google Scholar
[21]Michel, P. and Ullmo, E.Points de petite hauteur sur les courbes modulaires X0(N). Invent. Math. 131 (3) (1998), 645–674.Google Scholar
[22]Mochizuki, S.Correspondences on hyperbolic curves. J. Pure Appl. Algebra. 131 (3) (1998), 227–244.Google Scholar
[23]Moret-Bailly, L. Hauteurs et classes de Chern sur les surfaces arithmétiques. In: Séminaire sur les Pinceaux de Courbes Elliptiques (à la Recherche de Mordell effectif). Société de Mathématique de France Astérisque, vol. 183 (1990).Google Scholar
[24]Noohi, B. Foundations of topological stacks I. arXiv:math/0503247.Google Scholar
[25]Odlyzko, A. M.Lower bounds for discriminants of number fields. II. Tôhoku Math. J. 29 (2) (1977), 209–216.Google Scholar
[26]Raynaud, M.Hauteurs et isogénies. Astérisque. (127) (1985), 199–234. Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84).Google Scholar
[27]Shimizu, H.On zeta functions of quaternion algebras. Ann. of Math. (2), 81 (1965), 166–193.Google Scholar
[28]Shimura, G.Introduction to the arithmetic theory of automorphic functions. Publ. Math. Soc. Japan, vol, 11. (Princeton University Press, Princeton, NJ, 1994). Reprint of the 1971 original, Kanô Memorial Lectures, 1.Google Scholar
[29]Sijsling, J.Arithmetic (1; e)-curves and Belyi maps. Math. Comp. 81 (279) (2012), 1823–1855.Google Scholar
[30]Szpiro, L. editor. Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell. Société Mathématique de France, Paris, 1985. Papers from the seminar held at the École Normale Supérieure, Paris, (1983-84), Astérisque 127 (1985).Google Scholar
[31]Takeuchi, K.A characterisation of arithmetic Fuchsian groups. J. Math. Soc. Japan. 27 (4) (1975), 600–612.Google Scholar
[32]Takeuchi, K.Arithmetic Fuchsian groups with signature (1; e). J. Math. Soc. Japan. 35 (3) (1983), 381–407.Google Scholar
[33]Ullmo, E.Hauteur de Faltings de quotients de J0(N), discriminants d'algèbres de Hecke et congruences entre formes modulaires. Amer. J. Math. 122 (1) (2000), 83–115.Google Scholar
[34]Vakil, R. and Wickelgren, K.Universal covering spaces and fundamental groups in algebraic geometry as schemes. J. Théor. Nombres Bordeaux. 23 (2) (2011), 489–526.Google Scholar
[35]Voight, J.Enumeration of totally real number fields of bounded root discriminant. In: Algorithmic Number Theory. Lecture Notes in Comput. Sci. vol. 5011. (Springer, Berlin, 2008), 268–281.Google Scholar