Skip to main content
SpringerLink
Log in

Height functions for motives

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We define various height functions for motives over number fields. We compare these height functions with classical height functions on algebraic varieties, and also with analogous height functions for variations of Hodge structures on curves over \({\mathbb C}\). These comparisons provide new questions on motives over number fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Batyrev, V.V., Manin, Y.I.: On the number of rational points of bounded height on algebraic varieties. Math. Ann. 286, 27–43 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beilinson, A.: Higher regulators and values of \(L\)-functions. J. Sov. Math. 30, 2036–2070 (1985)

    Article  MATH  Google Scholar 

  3. Beilinson, A.: Height pairing between algebraic cycles. Contemp. Math. 67, 1–24 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bhatt, B., Morrow, M., Scholze, P.: Integral \(p\)-adic Hodge theory, preprint (2016), arXiv:1602.03148

  5. Bloch, S.: Height pairings for algebraic cycles. J. Pure Appl. Algebra 34, 119–145 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bloch, S., Kato, K.: L-Functions and Tamagawa Numbers of Motives. The Grothendieck Festschrift, vol. 1, Progress in Mathematics. Birkhäuser, pp. 333–400 (1990)

  7. Cattani, E., Kaplan, A., Schmid, W.: Degeneration of Hodge structures. Ann. Math. 123, 457–535 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. \(\check{\rm C}\)esnavi\(\check{\rm c}\)ius, K.: The \(A_{\text{inf}}\)-cohomology in the semi-stable case, preprint (2016)

  9. Deligne, P.: La conjecture de Weil, II. Publ. Math. IHES 52, 137–252 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73, 349–366 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Faltings, G.: Integral \(p\)-adic Hodge theory over very ramified valuation rings. J. Am. Math. Soc. 12, 117–144 (1999)

    Article  MATH  Google Scholar 

  12. Faltings, G.: Almost étale extensions. Astérisque 279, 185–270 (2002)

    MATH  Google Scholar 

  13. Fontaine, J.-M., Laffaille, G.: Construction de représentations \(p\)-adiques. Ann. Sci. École Norm. Sup. 15, 547–608 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gillet, H., Soulé, C.: Intersection sur les variétés d’Arakelov. C.R. Acad. Sci. Paris 299(Série I), 563–566 (1984)

    MathSciNet  MATH  Google Scholar 

  15. Gillet, H., Soulé, C.: Arithmetic intersection theory. Publ. Math. IHES 72, 93–174 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Green, W.: Heights in families of abelian varieties. Duke Math. J. 58, 617–632 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Griffiths, P.A.: Periods of integrals on algebraic manifolds, 12. Construction and properties of modular varieties. Am. J. Math. 90, 568–626 (1968)

    Article  MATH  Google Scholar 

  18. Griffiths, P.A.: Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping. Publ. Math. IHES 38, 125–180 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jannsen, U.: Mixed motives and algebraic \(K\)-theory. With appendices by S. Bloch and C. Schoen. Lecture Notes in Mathematics, vol. 1400 (1990)

  20. Kashiwara, M.: A study of variation of mixed Hodge structure. Publ. R.I.M.S. Kyoto Univ. 22, 991–1024 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kato, K.: p-adic period domains and toroidal partial compactifications, I. Kyoto J. Math. 51, 561–631 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kato, K.: Heights of motives. Proc. Japan Acad. Ser. A Math. Sci. 90, 49–53 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kato, K.: Heights of mixed motives, preprint (2013), arxiv:1306.5693

  24. Kato, K., Nakayama, C., Usui, S.: Classifying spaces of degenerating mixed Hodge structures, I. Adv. Stud. Pure Math. 54, 187–222 (2009)

    MATH  Google Scholar 

  25. Kato, K., Nakayama, C., Usui, S.: Classifying spaces of degenerating mixed Hodge structures, II. Kyoto J. Math. 51, 149–261 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kato, K., Nakayama, C., Usui, S.: Classifying spaces of degenerating mixed Hodge structures, III. J. Algebr. Geom. 22, 671–772 (2013)

    Article  MATH  Google Scholar 

  27. Kato, K., Usui, S.: Classifying spaces of degenerating polarized Hodge structures. In: Annals of Mathematics Studies, vol. 169 (2009)

  28. Kisin, M.: Crystalline representations and F-crystals. Algebraic geometry and number theory, Progr. Math. vol. 253. Birkhäuser, pp. 459–496 (2006)

  29. Koshikawa, T.: On heights of motives with semi-stable reduction, preprint (2015), arXiv:1505.01873

  30. Koshikawa, T.: Hodge bundles and heights of pure motives. Univ. of Chicago, Thesis (2016)

  31. Mochizuki, S.: The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories, preprint (1999)

  32. Mochizuki. S.: Inter-universal Teichmüller Theory I, II, III, IV, preprint (2012)

  33. Peters, C.: A criterion for flatness of Hodge bundles over curves and geometric applications. Math. Ann. 268, 1–19 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79, 101–218 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rapoport, M.: Non-Archimedean period domains. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994). Birkhäuser, pp. 423–434 (1995)

  36. Ribet, K.: Galois representations attached to eigenforms with Nebentypus. In: Modular Functions of One Variable, V, Lecture Notes in Math., vol. 601. Springer, pp. 17–51 (1977)

  37. Roberts, D.P.: Newforms with rational coefficients, preprint (2016), arXiv:1611.06967

  38. Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  39. Scholze, P.: Perfectoid spaces. Publ. Math. IHES 116, 245–313 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Scholze, P., Weinstein, J.: \(p\)-adic geometry. Lecture notes from course of P, Scholze at UC Berkeley in Fall (2014)

  41. Scholl, A.J.: Height pairings and special values of \(L\)-functions. Proc. Symp. Pure Math 55(Part 1), 571–598 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Schwarz, C.: Relative monodromy weight filtrations. Math. Z. 236, 11–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  43. Steenbrink, J., Zucker, S.: Variation of mixed Hodge structure. I. Invent. math. 80, 489–542 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tate, J.: Variation of the canonical height of a point depending on a parameter. Am. J. Math. 105, 287–294 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  45. Usui, S.: Variation of mixed Hodge structures arising from family of logarithmic deformations. II. Classifying space. Duke Math. J. 51, 851–875 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  46. Vojta, P.: A more general abc conjecture. Int. Math. Res. Not. 21, 1103–1116 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  47. Vojta, P.: Diophantine approximation and Nevanlinna theory. In: Arithmetic Geometry, Cetraro, Italy 2007, Lecture Notes in Mathematics, vol. 2009, pp. 111–230. Springer (2011)

  48. Zucker, S.: Hodge theory with degenerating coefficients: L\(^2\)-cohomology in the Poincaré metric. Ann. Math. 109, 415–476 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Chicago, Chicago, IL, USA

    Kazuya Kato

Authors
  1. Kazuya Kato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazuya Kato.

Additional information

Dedicated to Professor Alexander Beilinson on his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kato, K. Height functions for motives. Sel. Math. New Ser. 24, 403–472 (2018). https://doi.org/10.1007/s00029-017-0376-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-017-0376-9

Mathematics Subject Classification

Search

Navigation