Advertisement

Inventiones mathematicae

, Volume 209, Issue 2, pp 481–539 | Cite as

New explicit formulas for Faltings’ delta-invariant

  • Robert WilmsEmail author
Article

Abstract

In this paper we give new explicit formulas for Faltings’ \(\delta \)-invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for \(\delta \) only in terms of the genus and an explicit upper bound for the Arakelov–Green function in terms of \(\delta \). Furthermore, we give a canonical extension of \(\delta \) and the Zhang–Kawazumi invariant \(\varphi \) to the moduli space of indecomposable principally polarised complex abelian varieties.

Mathematics Subject Classification

14G40 14H42 14H55 14K25 11G30 

Notes

Acknowledgements

The contents of this paper largely coincide with my Ph.D. thesis. I would like to thank my advisor Gerd Faltings for introducing me into Arakelov theory and for his suggestion to study the \(\delta \)-invariant. I also would like to thank Rafael von Känel and Robin de Jong for useful discussions and Michael Rapoport for a remark on an early version of this paper.

References

  1. 1.
    Arakelov, S.Y.: Intersection theory of divisors on an arithmetic surface. Izv. Akad. USSR 8(6), 1167–1180 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Autissier, P.: An upper bound for the theta function (2015). Preprint. https://www.math.u-bordeaux.fr/~pautissi/Theta. Accessed 27 Oct 2015
  3. 3.
    Birkenhake, C., Lange, H.: Complex Abelian Varieties. Grundlehren der Mathematischen Wissenschaften, vol. 302. Springer, Berlin (2004)Google Scholar
  4. 4.
    Bost, J.-B., Mestre, J.-F., Moret-Bailly, L.: Sur le calcul explicite des “classes de Chern” des surfaces arithmétiques de genre 2. In: Séminaire sur les Pinceaux de Courbes Elliptiques, Astérisque, vol. 183, pp. 69–105 (1990)Google Scholar
  5. 5.
    Bost, J.-B.: Fonctions de Green-Arakelov, fonctions thêta et courbes de genre 2. C. R. Acad. Sci. Paris Sér. I Math. 305(14), 643–646 (1987)Google Scholar
  6. 6.
    Deligne, P.: Le déterminant de la cohomologie. In: Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985). Contemporary Mathematics, vol. 67, pp. 387–424 (1987)Google Scholar
  7. 7.
    Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry—Methods and Applications—Part II. The Geometry and Topology of Manifolds. Graduate Texts in Mathematics, vol. 104. Springer, New York (1985)Google Scholar
  8. 8.
    Edixhoven, B., Couveignes, J.-M.: Computational Aspects of Modular Forms and Galois Representations. Annals of Mathematics Studies, vol. 176. Princeton University Press, Princeton (2011)Google Scholar
  9. 9.
    Elkik, R.: Fibrés d’intersections et intégrales de classes de Chern. Ann. Sci. Ecole Norm. Sup. 22(2), 195–226 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Faltings, G.: Calculus on arithmetic surfaces. Ann. Math. 119, 387–424 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Farkas, H.M., Kra, I.: Riemann Surfaces. Graduate Texts in Mathematics, vol. 71. Springer, New York (1980)Google Scholar
  12. 12.
    Guàrdia, J.: Analytic invariants in Arakelov theory for curves. C.R. Acad. Sci. Paris Ser. I 329, 41–46 (1999)Google Scholar
  13. 13.
    Guàrdia, J.: Jacobian nullwerte and algebraic equations. J. Algebra 253(1), 112–132 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hain, R., Looijenga, E.: Mapping class groups and moduli spaces of curves. In: Algebraic Geometry-Santa Cruz 1995. Proceedings of Symposia in Pure Mathematics, vol. 62, pp. 97–142 (1997)Google Scholar
  15. 15.
    Hain, R., Reed, D.: Geometric proofs of some results of Morita. J. Algebraic Geom. 10(2), 199–217 (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hain, R., Reed, D.: On the Arakelov geometry of moduli spaces of curves. J. Differ. Geom. 67(2), 195–228 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Igusa, J.-I.: Theta Functions. Grundlehren der mathematischen Wissenschaften, vol. 194. Springer, New York (1972)Google Scholar
  18. 18.
    Javanpeykar, A.: Polynomial bounds for Arakelov invariants of Belyi curves. With an appendix by Peter Bruin. Algebra Number Theory 8(1), 89–140 (2014)Google Scholar
  19. 19.
    de Jong, R.: Arakelov invariants of Riemann surfaces. Doc. Math. 10, 311–329 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    de Jong, R.: Faltings’ delta-invariant of a hyperelliptic Riemann surface. In: van der Geer, G., Moonen, B., Schoof, R. (eds.) Number Fields and Function Fields-Two Parallel Worlds, Progress in Mathematics, vol. 239, pp. 223–236. Birkhäuser, Basel (2005)Google Scholar
  21. 21.
    de Jong, R.: Explicit Mumford isomorphism for hyperelliptic curves. J. Pure Appl. Algebra 208, 1–14 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    de Jong, R.: Gauss map on the theta divisor and Green’s functions. In: Edixhoven, B., van der Geer, G., Moonen, B. (eds.) Modular Forms on Schiermonnikoog, pp. 67–78. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  23. 23.
    de Jong, R.: Theta functions on the theta divisor. Rocky Mt. J. Math. 40, 155–176 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    de Jong, R.: Second variation of Zhang’s \(\lambda \)-invariant on the moduli space of curves. Am. J. Math. 135, 275–290 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    de Jong, R.: Asymptotic behavior of the Kawazumi–Zhang invariant for degenerating Riemann surfaces. Asian J. Math. 18, 507–524 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    de Jong, R.: Torus bundles and 2-forms on the universal family of Riemann surfaces. In: Papadopoulos, A. (ed.) Handbook of Teichmüller Theory, vol. VI, pp. 195–227. EMS Publishing House, Zurich (2016)Google Scholar
  27. 27.
    Jorgenson, J., Kramer, J.: Bounds on canonical Green’s functions. Compos. Math. 142(3), 679–700 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jorgenson, J., Kramer, J.: Bounds on Faltings’s delta function through covers. Ann. Math. (2) 170(1), 1–43 (2009)Google Scholar
  29. 29.
    Jorgenson, J.: Asymptotic behavior of Faltings’s delta function. Duke Math. J. 61, 221–254 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    von Känel, R.: On Szpiro’s discriminant conjecture. Int. Math. Res. Not. 16, 4457–4491 (2014)MathSciNetzbMATHGoogle Scholar
  31. 31.
    von Känel, R.: Integral points on moduli schemes of elliptic curves. Trans. Lond. Math. Soc. 1(1), 85–115 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kawazumi, N.: Johnson’s homomorphisms and the Arakelov–Green function (2008). Preprint. arXiv:0801.4218. Accessed 09 Dec 2014
  33. 33.
    Lockhart, P.: On the discriminant of a hyperelliptic curve. Trans. Am. Soc. 342(2), 729–752 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mumford, D.: Tata Lectures on Theta I. Progress in Mathematics, vol. 28. Birkhäuser, Basel (1983)Google Scholar
  35. 35.
    Mumford, D.: Tata Lectures on Theta II. Progress in Mathematics, vol. 43. Birkhäuser, Basel (1984)Google Scholar
  36. 36.
    Pioline, B.: A theta lift representation for the Kawazumi–Zhang and Faltings invariants of genus-two Riemann surfaces. J. Number Theory 163, 520–541 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Powell, J.: Two theorems on the mapping class group of a surface. Proc. Am. Math. Soc. 68(3), 347–350 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Rosenhain, G.: Mémoire sur les fonctions de deux variables et à quatre périodes qui sont les inverses des intégrales ultra-elliptiques de la première classe. Mémoires des savants étrangers 11, 362–468 (1851)Google Scholar
  39. 39.
    Schlichenmaier, M.: An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Theoretical and Mathematical Physics. Springer, Berlin (2007)Google Scholar
  40. 40.
    Szpiro, L.: Degrés, intersections, hauteurs. Astérisque 127, 11–28 (1985)zbMATHGoogle Scholar
  41. 41.
    Wentworth, R.: The asymptotics of the Arakelov–Green’s function and Faltings’ delta invariant. Commun. Math. Phys. 137, 427–459 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhang, S.: Heights and reductions of semi-stable varieties. Compos. Math. 104(1), 77–105 (1996)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Zhang, S.: Gross–Schoen cycles and dualising sheaves. Invent. Math. 179, 1–73 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany

Personalised recommendations