Singular moduli of higher level and special cycles

  • Stephan Ehlen
Open Access


We describe the complex multiplication (CM) values of modular functions for \(\Gamma _0(N)\) whose divisor is given by a linear combination of Heegner divisors in terms of special cycles on the stack of CM elliptic curves. In particular, our results apply to Borcherds products of weight \(0\) for \(\Gamma _0(N)\). By working out explicit formulas for the special cycles, we obtain the prime ideal factorizations of such CM values.


Complex multiplication CM values Special cycles Borcherds products 

Mathematics Subject Classification

Primary 11F03 Secondary 11G15 



This article contains some results that were contained in my thesis [7]. First and foremost I would like to thank my advisor Jan Hendrik Bruinier for his constant support and encouragement. I also thank Claudia Alfes and Tonghai Yang for their help. I also thank Benjamin Howard for helpful discussions and in particular for helping me with the proof of Proposition 2.5. Moreover, the helpful comments of the anonymous referee are greatly appreciated.

Role of funding source

This work is partially supported by DFG grant BR-2163/4-1.

Compliance with ethical guidelines

Competing interests The author declares that he has no competing interests.


  1. 1.
    Weber, H.: Algebra, Dritter Band, 3rd edn. Chelsea Publishing Company, New York (1961)Google Scholar
  2. 2.
    Zagier, D.B.: Traces of singular moduli. In: Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998). Int. Press Lect. Ser., vol. 3, pp. 211–244. International Press, Somerville (2002)Google Scholar
  3. 3.
    Gross, B.H., Zagier, D.B.: On singular moduli. J. Reine Angew. Math. 355, 191–220 (1985). doi: 10.1016/j.jnt.2006.04.009 MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dorman, D.R.: Special values of the elliptic modular function and factorization formulae. J. Reine Angew. Math. 383, 207–220 (1988). doi: 10.1515/crll.1988.383.207 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gross, B.H., Kohnen, W., Zagier, D.B.: Heegner points and derivatives of L-series. II. Math. Ann. 278(1–4), 497–562 (1987). doi: 10.1007/BF01458081 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bruinier, J.H., Yang, T.: Faltings heights of CM cycles and derivatives of L-functions. Invent. Math. 177(3), 631–681 (2009). doi: 10.1007/s00222-009-0192-8 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ehlen, S.: CM Values of Regularized Theta Lifts. TU Darmstadt, Darmstadt (2013).
  8. 8.
    Ehlen, S.: CM values of regularized theta lifts (preprint) (2015)Google Scholar
  9. 9.
    Kudla, S.S., Rapoport, M., Yang, T.: On the derivative of an Eisenstein series of weight one. Int. Math. Res. Notices 7, 347–385 (1999). doi: 10.1155/S1073792899000185 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Borcherds, R.E.: The Gross–Kohnen–Zagier theorem in higher dimensions. Duke Math. J. 97(2), 219–233 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bruinier, J.H.: On the converse theorem for Borcherds products. ArXiv:1210.4821 (e-prints) (2012).
  12. 12.
    Kudla, S.S., Rapoport, M., Yang, T.: Derivatives of Eisenstein series and Faltings heights. Compos. Math. 140(4), 887–951 (2004). doi: 10.1112/S0010437X03000459 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Koblitz, N.: Introduction to elliptic curves and modular forms, 2nd edn. In: Graduate Texts in Mathematics, vol. 97, p. 248. Springer, New York (1993)Google Scholar
  14. 14.
    Miyake, T.: Modular forms, English edn. In: Springer Monographs in Mathematics, p. 335. Springer, Berlin (2006). Translated from the 1976 Japanese original by Yoshitaka MaedaGoogle Scholar
  15. 15.
    Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and q-series. In: CBMS Regional Conference Series in Mathematics, vol. 102, p. 216. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2004)Google Scholar
  16. 16.
    Stein, W.: Modular forms, a computational approach. In: Graduate Studies in Mathematics, vol. 79, p. 268. American Mathematical Society, Providence (2007). With an appendix by Paul E. GunnellsGoogle Scholar
  17. 17.
    Shimura, G.: Introduction to the arithmetic theory of automorphic functions. In: Publications of the Mathematical Society of Japan, vol. 11, p. 271. Princeton University Press, Princeton (1994). Reprint of the 1971 original, Kanô Memorial Lectures, 1Google Scholar
  18. 18.
    Katz, N.M., Mazur, B.: Arithmetic moduli of elliptic curves. In: Annals of Mathematics Studies, vol. 108, p. 514. Princeton University Press, Princeton (1985)Google Scholar
  19. 19.
    Silverman, J.H.: The arithmetic of elliptic curves, 2nd edn. In: Graduate Texts in Mathematics, vol. 106, p. 513. Springer, Dordrecht (2009). doi: 10.1007/978-0-387-09494-6
  20. 20.
    Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. In: Graduate Texts in Mathematics, vol. 151, p. 525. Springer, New York (1994). doi: 10.1007/978-1-4612-0851-8
  21. 21.
    Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg 14(1), 197–272 (1941). doi: 10.1007/BF02940746 MathSciNetCrossRefGoogle Scholar
  22. 22.
    Howard, B.: Rankin-Selberg L-functions and cycles on unitary Shimura varieties. (2013)
  23. 23.
    Lan, K.-W.: Arithmetic compactifications of PEL-type shimura varieties, p. 1077. Ph.D. Thesis, Harvard University. ProQuest LLC, Ann Arbor (2008)Google Scholar
  24. 24.
    Bruinier, J.H., Howard, B., Yang, T.: Heights of Kudla-Rapoport divisors and derivatives of L-functions. Invent. Math. ArXiv:1303.0549 (e-prints) (2013, to appear)
  25. 25.
    Howard, B.: Moduli spaces of CM elliptic curves. (2012)
  26. 26.
    Conrad, B.: Gross-Zagier revisited. In: Heegner Points and Rankin L-series. Math. Sci. Res. Inst. Publ., vol. 49, pp. 67–163. Cambridge University Press, Cambridge (2004). doi: 10.1017/CBO9780511756375.006. With an appendix by W. R. Mann
  27. 27.
    Kudla, S.S., Yang, T.: On the pullback of an arithmetic theta function. Manuscripta Math. 140(3–4), 393–440 (2013). doi: 10.1007/s00229-012-0569-7 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Serre, J.-P.: A course in arithmetic. In: Graduate Texts in Mathematics, vol. 7, p. 115. Springer, New York (1973). Translated from the FrenchGoogle Scholar
  29. 29.
    Vistoli, A.: Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97(3), 613–670 (1989). doi: 10.1007/BF01388892 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ehlen, S.: Vector valued theta functions associated with binary quadratic forms. ArXiv:1505.02693 (preprint) (2015)
  31. 31.
    Dorman, D.R.: Global orders in definite quaternion algebras as endomorphism rings for reduced CM elliptic curves. Théorie des Nombres (Quebec. PQ, 1987), pp. 108–116. de Gruyter, Berlin (1989)Google Scholar
  32. 32.
    Zagier, D.B.: Zetafunktionen und Quadratische Köorper, p. 144. Springer, Berlin (1981). Eine Einführung in die höhere Zahlentheorie (An introduction to higher number theory). Hochschultext (University Text)Google Scholar
  33. 33.
    Serre, J.-P.: Local fields. In: Graduate Texts in Mathematics, vol. 67, p. 241. Springer, New York (1979). Translated from the French by Marvin Jay GreenbergGoogle Scholar
  34. 34.
    Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math. 2(88), 492–517 (1968)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular Functions of One Variable, II (Proceedings of International Summer School, University of Antwerp, Antwerp, 1972), pp. 143–316349. Springer, Berlin (1973)Google Scholar
  36. 36.
    Gross, B.H., Zagier, D.B.: Heegner points and derivatives of L-series. Invent. Math. 84(2), 225–320 (1986). doi: 10.1007/BF01388809 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Conrad, B.: Arithmetic moduli of generalized elliptic curves. J. Inst. Math. Jussieu 6(2), 209–278 (2007). doi: 10.1017/S1474748006000089 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Birch, B.J.: Heegner points of elliptic curves. In: Symposia Mathematica, vol. XV (Convegno di Strutture in Corpi Algebrici, INDAM, Rome, 1973), pp. 441–445. Academic Press, London (1975)Google Scholar
  39. 39.
    Gross, B.H.: Heegner points on X0(N). In: Modular Forms (Durham, 1983). Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., pp. 87–105. Horwood, Chichester (1984)Google Scholar
  40. 40.
    Diamond, F., Im, J.: Modular forms and modular curves. In: Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994). CMS Conference Proceedings, vol. 17, pp. 39–133. American Mathematical Society, Providence (1995)Google Scholar
  41. 41.
    Borcherds, R.E.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132(3), 491–562 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Bruinier, J., Ono, K.: Heegner divisors, L-functions and harmonic weak Maass forms. Ann. Math. 2 172(3), 2135–2181 (2010). doi: 10.4007/annals.2010.172.2135 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Borcherds, R.E.: Correction to: “The Gross–Kohnen–Zagier theorem in higher dimensions” [Duke Math. J. 97 (1999), no. 2, 219–233]. Duke Math. J. 105(1), 183–184 (2000). doi: 10.1215/S0012-7094-00-10519-4 MathSciNetCrossRefGoogle Scholar
  44. 44.
    Bruinier, J.H., Ehlen, S., Freitag, E.: Lattices with many Borcherds products. ArXiv:1408.4148 (e-prints) (2014)

Copyright information

© Ehlen. 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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