Skip to main content
SpringerLink
Log in

Kronecker’s first limit formula, revisited

  • Research
  • Published:
Research in the Mathematical Sciences Aims and scope Submit manuscript

Abstract

We give some new applications of Kronecker’s first limit formula to real quadratic fields. In particular, we give a surprising geometrical relationship between the CM points associated with two imaginary quadratic fields with discriminants d and \(d^{\prime }\) and certain winding number functions coming from the closed geodesics associated with the real quadratic field of discriminant \(d^{\prime }d\).

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Asai, T., Kaneko, M., Ninomiya, H.: Zeros of certain modular functions and an application. Comment. Math. Univ. St. Paul 46(1), 93–101 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Berger, A.: Sur une application des nombres des classes des formes quadratiques binaires pour un déterminant négatif. Nova Acta Soc. Sci. Upsaliensis 3(12), 29–31 (1883)

    Google Scholar 

  3. Birch, B.J.: Diophantine analysis and modular functions. In: 1969 Algebraic Geometry (International Colloqium, Tata Institute Fundamental Research, Bombay) pp. 35–42. Oxford University Press, London (1968)

  4. Birch, B.J.: Heegner’s Proof. Arithmetic of L-Functions. IAS/Park City Mathematical Series, 18, pp. 281–291. American Mathematical Society, Providence, RI (2011)

    MATH  Google Scholar 

  5. Calegari, D.S.C.L.: MSJ Memoirs, 20. Mathematical Society of Japan, Tokyo (2009)

    Google Scholar 

  6. Calegari, D., Louwsma, J.: Immersed surfaces in the modular orbifold. Proc. Am. Math. Soc. 139, 2295–2308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chowla, S., Selberg, A.: On Epstein’s zeta function (I). Proc. Nat. Acad. Sci. U.S.A. 35, 371–374 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  8. Selberg, A., Chowla, S.: On Epstein’s zeta-function. J. Reine Angew. Math. 227, 86–110 (1967)

    MathSciNet  MATH  Google Scholar 

  9. Cohen, H., Lenstra Jr., H.W.: Heuristics on Class Groups. Number theory (New York, 1982). Lecture Notes in Mathematics, 1052, pp. 26–36. Springer, Berlin (1984)

    Google Scholar 

  10. Deninger, C.: On the analogue of the formula of Chowla and Selberg for real quadratic fields. J. Reine Angew. Math. 351, 171–191 (1984)

    MathSciNet  MATH  Google Scholar 

  11. Deuring, M.F.: On Epstein’s zeta function. Ann. Math. (2) 38(3), 585–593 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deuring, M.: Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins. Invent. Math. 5(3), 169–179 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  13. D’Hoker, E., Phong, D.H.: On determinants of Laplacians on Riemann surfaces. Commun. Math Phys. 104, 537–545 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Duke, W., Imamoḡlu, Ö., Tóth, Á.: Cycle integrals of the j-function and mock modular forms. Ann. Math. (2) 173(2), 947–981 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Duke, W., Imamoḡlu, Ö., Tóth, Á.: Geometric invariants for real quadratic fields. Ann. Math. 184(2), 94–990 (2016)

    MathSciNet  Google Scholar 

  16. Duke, W., Imamoḡlu, Ö., Tóth, Á.: Modular cocycles and linking numbers. Duke Math. J. 166(6), 1179–1210 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fay, John: Analytic torsion and Prym differentials. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State University New York, Stony Brook, NY, 1978), pp. 107–122. Annals in Mathematical Studies, 97. Princeton University Press, Princeton, NJ (1981)

  18. Ghys, E.: Knots and Dynamics. International Congress of Mathematicians, vol. I, pp. 247–277. European Mathematical Society, Zürich (2007)

    MATH  Google Scholar 

  19. Rademacher, H., Grosswald, E.: Dedekind Sums, The Carus Mathematics Monographs, MAA (1972)

  20. Hammond, W.F., Hirzebruch, F.: L-series, modular imbeddings, and signatures. Math. Ann. 204, 263–270 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hecke, E.: Über die Kroneckersche Grenzformell für reelle quadratische Körper und die Klassenzhal relative-Abelscher Körper, Verhandlungen der Naturforschenden Gesellschaft in Basel Bd. 28, S.363–372 = Werke 10, pp. 198–207 (1917)

  22. Hecke, E.: Bestimmung der Klassenzahl einer neuen Reihe von algebraischen Zahlkörpern, Nachrichten der K. Gesellschaft der Wissenschafter zu Göttingen. Mathematische-physikalische Klasse, S.1–23= Werke 15, pp. 290–312 (1921)

  23. Hecke, E.: Darstellung von Klassenzahl als Peioden von Integralen3. Gattung aus dem Gebiet der elliptischen Modulfunktionen, Abhandlungen aus dem Mathematichen Seminar der Hamburgischen Universität Bd. 4, 1925, S.211–223=Werke 21, pp. 405–417

  24. Heegner, K.: Diophantische analysis und modulfunktionen. Math. Z. 56(3), 227–253 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  25. G. Herglotz, Über die Kroneckersche Grenzformel für reele, quadratische Körper. I, II, Leipziger Ber. 75 (1923), pp. 3–14 and pp. 31–37; Ges. Schriften, pp. 466–484

  26. Hirzebruch, F.E.P.: Hilbert modular surfaces. Enseign. Math. (2) 19, 183–281 (1973)

    MathSciNet  MATH  Google Scholar 

  27. Hirzebruch, F., Zagier, D.: Classification of Hilbert modular surfaces. In: Complex Analysis and Algebraic Geometry, pp. 43–77. Collected Papers II. Iwanami Shoten, Tokyo (1977)

  28. Hurwitz, A.: Einige Eigenschaften der Dirichlet’schen Funktionen \(F(s)=\sum (D/n)1/(n^s)\), die bei der Bestimmung der Klassenanzahlen Binärer quadratischer Formen auftreten. Z. Math. Phys. 27, 86–101 (1882)

    Google Scholar 

  29. Katok, S.: Coding of closed geodesics after Gauss and Morse. Geom. Dedicata 63(2), 123–145 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kronecker, Werke IV Chelsea pp. 221–225 (1929)

  31. Lagarias, J.C., Rhoades, R.C.: Polyharmonic maass forms for \(\text{ PSL }(2,{{\mathbb{Z}}})\). Ramanujan J. 41(1–3), 191–232 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Landau, E.: Über die zu einem algebraischen Zahlkörper gehörige Zetafunction und die Ausdehnung der Tschebyschefschen Primzahltheorie auf das Problem der Verteilung der Primideale J. reine angew. Math. (Crelles J.) 125, 64–188 (1902). (Collected Works Vol. I, pp. 201–326)

    MathSciNet  Google Scholar 

  33. Lang, S.: Elliptic Functions. With an Appendix by J. Tate, 2nd edn. Graduate Texts in Mathematics, 112. Springer, New York (1987)

    Book  MATH  Google Scholar 

  34. Lerch, M.: Sur quelques formules relatives du nombre des classes. Bull. Sci. Math. 21, 290–304 (1897)

    MATH  Google Scholar 

  35. Muzaffar, H., Williams, K.S.: A restricted Epstein zeta function and the evaluation of some definite integrals. Acta Arithm. 104(1), 23–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Meyer, C.: Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern. (German) Akademie-Verlag, Berlin (1957)

  37. Ramachandra, K.: Some applications of Kronecker’s limit formulas. Ann. Math. 2(80), 104–148 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math. 2(98), 154–177 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  39. Sarnak, P.: Special Values of Selberg’s Zeta-Function. Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), pp. 457–465. Academic Press, Boston, MA (1989)

    MATH  Google Scholar 

  40. Sarnak, P.: Determinants of Laplacians; Heights and Finiteness. Analysis, et cetera, pp. 601–622. Academic Press, Boston, MA (1990)

    Google Scholar 

  41. Sarnak, P.: Reciprocal Geodesics. In: Analytic Number Theory, Clay Math. Proceedings, 7, pp. 217–237. American Mathemaical Society, Providence, RI (2007)

  42. Shintani, T.: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24(1), 167–199 (1977)

    MathSciNet  MATH  Google Scholar 

  43. Shintani, T.: A proof of the classical Kronecker limit formula. Tokyo J. Math. 3(2), 191–199 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  44. Siegel, C.L.: Advanced Analytic Number Theory. Tata, Bombay (1980)

    MATH  Google Scholar 

  45. Siegel, C.L.: Bernoullische Polynome und quadratische Zahlkörper, Collected Papers IV, pp. 9–40 (1968)

  46. Stark, H.M.: Class-number problems in quadratic fields. Actes du Congrés International des Mathématiciens (Nice, 1970), Tome 1, pp. 511–518. Gauthier-Villars, Paris (1971)

  47. Stark, H.M.: Values of L-functions at s=1. I. L-functions for quadratic forms. Adv. Math. 7, 301–343 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stark, H.: The origin of the “Stark conjectures”. In: Arithmetic of L-Functions. IAS/Park City Mathematical Series, 18, pp. 33–44. Americal Mathematical Society, Providence, RI (2011)

  49. Vlasenko, M., Zagier, D.: Higher Kronecker limit formulas for real quadratic fields. J. Reine Angew Math. 679, 23–64 (2013)

    MathSciNet  MATH  Google Scholar 

  50. Weber, H.M.: Elliptische functionen und algebraische zahlen. Vieweg und Sohn, Braunschweig (1891)

    MATH  Google Scholar 

  51. Weber, H.M.: Lehrbuch der Algebra, vol. 3, 3rd edn. Chelsea, New York (1961)

    Google Scholar 

  52. Weil, A.: Elliptic Functions According to Eisenstein and Kronecker. Reprint of the 1976 original. Classics in Mathematics. Springer, Berlin (1999)

  53. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions, 4th edn. Reprinted Cambridge University Press, New York (1962)

  54. Zagier, D.: A Kronecker limit formula for real quadratic fields. Math. Ann. 213, 153–184 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zagier, D.: Nombres de classes et fractions continues. J. Arithm. Bordeaux Astérisque 24–25, 81–97 (1975)

    MathSciNet  MATH  Google Scholar 

  56. Zagier, D.B.: Zetafunktionen und quadratische Körper (German) [Zeta functions and quadratic fields]. Eine Einführung in die höhere Zahlentheorie [An introduction to higher number theory]. Hochschultext [University Text]. Springer, Berlin (1981)

Download references

Author's contributions

Acknowlegements

Duke and Tóth are grateful to FIM of ETH Zürich for its generous continued support of our joint research. Duke’s research on this paper was supported by NSF Grant DMS 1701638, the Simons Foundation and the Mathematisches Forschungsinstitut Oberwolfach. Á. Tóth is supported by NKFIH (National Research, Development and Innovation Office) Grant ERC_HU_15 118946.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Author information

Authors and Affiliations

  1. Mathematics Department, UCLA, Box 951555, Los Angeles, CA, 90095-1555, USA

    W. Duke

  2. Mathematics Department, ETH, 8092, Zurich, Switzerland

    Ö. Imamoḡlu

  3. Eotvos Lorand University, South Building Room 3.207, Budapest, Hungary

    Á. Tóth

Authors
  1. W. Duke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ö. Imamoḡlu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Á. Tóth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. Duke.

Additional information

To Don Zagier, with admiration

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duke, W., Imamoḡlu, Ö. & Tóth, Á. Kronecker’s first limit formula, revisited. Res Math Sci 5, 20 (2018). https://doi.org/10.1007/s40687-018-0138-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-018-0138-0

Search

Navigation