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Modularity of generating series of winding numbers

  • Jan H. Bruinier
  • Jens Funke
  • Özlem ImamoḡluEmail author
  • Yingkun Li
Research
  • 170 Downloads
Part of the following topical collections:
  1. Modular Forms are Everywhere: Celebration of Don Zagier’s 65th Birthday

Abstract

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

Notes

Author's contributions

Acknowlegements

Most of this work was done during visits by J.F. to TU Darmstadt, by J.F. and Y.L. to FIM in Zürich in 2017. We thank these organizations for their supports. We also thank the referees for careful readings and helpful comments and corrections. J.B and Y.L. are supported by the DFG Grant BR-2163/4-2 and the LOEWE research unit USAG. Y.L. is also partially supported by an NSF postdoctoral fellowship.

References

  1. 1.
    Alfes, C., Ehlen, S.: Twisted traces of CM values of weak Maass forms. J. Number Theory 133(6), 1827–1845 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borcherds, R.E.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132(3), 491–562 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruinier, J., Ono, K.: Heegner divisors, \(L\)-functions and harmonic weak Maass forms. Ann. Math. (2) 172(3), 2135–2181 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bruinier, J.H., Funke, J.: Traces of CM values of modular functions. J. Reine Angew. Math. 594, 1–33 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bruinier, J.H., Schwagenscheidt, M.: Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products. J. Algebra 478, 38–57 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dabholkar, A., Murthy, S., Zagier, D.: Quantum black holes, wall crossing, and mock modular forms (2012). arXiv:1208.4074
  8. 8.
    Funke, J.: Heegner divisors and nonholomorphic modular forms. Compos. Math. 133(3), 289–321 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Funke, J., Kudla, S.: Mock modular forms and geometric theta functions for indefinite quadratic forms. J. Phys. A Math. Theor. 50, 404001 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Funke, J., Kudla, S.: Theta integrals and generalized error functions, II (2017). arXiv:1708.02969
  11. 11.
    Funke, J., Millson, J.: Cycles in hyperbolic manifolds of non-compact type and Fourier coefficients of Siegel modular forms. Manuscr. Math. 107(4), 409–444 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Griffiths, P.A.: Introduction to Algebraic Curves. Translations of Mathematical Monographs, vol. 76, American Mathematical Society, Providence (1989). Translated from the Chinese by Kuniko WeltinGoogle Scholar
  13. 13.
    Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives of \(L\)-series. II. Math. Ann. 278(1–4), 497–562 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hirzebruch, F., Zagier, D.: Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus. Invent. Math. 36, 57–113 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kudla, S.: Theta integrals and generalized error functions. Manuscr. Math. 155, 303–333 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kudla, S.S., Millson, J.J.: Geodesic cyclics and the Weil representation. I. Quotients of hyperbolic space and Siegel modular forms. Compos. Math. 45(2), 207–271 (1982)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kudla, S.S., Millson, J.J.: The theta correspondence and harmonic forms. II. Math. Ann. 277(2), 267–314 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kudla, S.S., Millson, J.J.: Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables. Inst. Hautes Études Sci. Publ. Math. 71, 121–172 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Scholl, A.J.: Fourier coefficients of Eisenstein series on noncongruence subgroups. Math. Proc. Camb. Philos. Soc. 99(1), 11–17 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shimura, G.: On modular forms of half integral weight. Ann. Math. (2) 97, 440–481 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shintani, T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58, 83–126 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Skoruppa, N.-P.: Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms. Invent. Math. 102(3), 501–520 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Voight, J.: Shimura Curve Computations. Arithmetic Geometry, Clay Mathematics Proceedings, vol. 8, pp. 103–113. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  24. 24.
    Waldschmidt, M.: Nombres transcendants et groupes algébriques. Astérisque, vol. 69. Société Mathématique de France, Paris (1979). With appendices by Daniel Bertrand and Jean-Pierre Serre, With an English summaryGoogle Scholar
  25. 25.
    Washington, L.C.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83, 2nd edn. Springer, New York (1997)CrossRefGoogle Scholar
  26. 26.
    Wüstholz, G.: Zum Periodenproblem. Invent. Math. 78(3), 381–391 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zagier, D.: Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann). Astérisque (2009), no. 326, Exp. No. 986, vii–viii, 143–164 (2010). Séminaire Bourbaki, vol. 2007/2008Google Scholar
  28. 28.
    Zwegers, S.P.: Mock \(\theta \)-Functions and Real Analytic Modular Forms, \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000). Contemporary Mathematics, vol. 291, pp. 269–277. American Mathematical Society, Providence (2001)Google Scholar
  29. 29.
    Zwegers, S.P.: Mock theta functions. Proefschrift Universiteit Utrecht, Thesis (Ph.D.), Universiteit Utrecht (2002)Google Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  • Jan H. Bruinier
    • 1
  • Jens Funke
    • 2
  • Özlem Imamoḡlu
    • 3
    Email author
  • Yingkun Li
    • 1
  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Department of Mathematical SciencesUniversity of DurhamDurhamUK
  3. 3.Departement MathematikETH ZürichZurichSwitzerland

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