Cycle integrals of the $j$-function and mock modular forms

Abstract

In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are given in terms of cycle integrals of the modular $j$-function. Their shadows are weakly holomorphic forms of weight 3/2. These new mock modular forms occur as holomorphic parts of weakly harmonic Maass forms. We also construct a generalized mock modular form of weight 1/2 having a real quadratic class number times a regulator as a Fourier coefficient. As an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers. The Fourier coefficients of these modular integrals are given in terms of cycle integrals of modular functions. Such a modular integral can be interpreted in terms of a Shimura-type lift of a mock modular form of weight 1/2 and yields a real quadratic analogue of a Borcherds product.

  • [AKN] T. Asai, M. Kaneko, and H. Ninomiya, "Zeros of certain modular functions and an application," Comment. Math. Univ. St. Paul, vol. 46, iss. 1, pp. 93-101, 1997.
    @article {AKN, MRKEY = {1448475},
      AUTHOR = {Asai, Tetsuya and Kaneko, Masanobu and Ninomiya, Hirohito},
      TITLE = {Zeros of certain modular functions and an application},
      JOURNAL = {Comment. Math. Univ. St. Paul},
      FJOURNAL = {Commentarii Mathematici Universitatis Sancti Pauli},
      VOLUME = {46},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {93--101},
      ISSN = {0010-258X},
      CODEN = {COMAAC},
      MRCLASS = {11F03 (11F11)},
      MRNUMBER = {1448475},
      MRREVIEWER = {Michael C. Berg},
      ZBLNUMBER = {0907.11013},
      }
  • [Bo1] Go to document R. E. Borcherds, "Automorphic forms on ${ O}_{s+2,2}({\bf R})$ and infinite products," Invent. Math., vol. 120, iss. 1, pp. 161-213, 1995.
    @article {Bo1, MRKEY = {1323986},
      AUTHOR = {Borcherds, Richard E.},
      TITLE = {Automorphic forms on {${\rm O}\sb {s+2,2}({\bf R})$} and infinite products},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {120},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {161--213},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F55 (11F11 11F22 17B67 20G20)},
      MRNUMBER = {1323986},
      MRREVIEWER = {Rainer Schulze-Pillot},
      DOI = {10.1007/BF01241126},
      ZBLNUMBER = {0932.11028},
      }
  • [BO] Go to document K. Bringmann and K. Ono, "Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series," Math. Ann., vol. 337, iss. 3, pp. 591-612, 2007.
    @article {BO, MRKEY = {2274544},
      AUTHOR = {Bringmann, Kathrin and Ono, Ken},
      TITLE = {Arithmetic properties of coefficients of half-integral weight {M}aass-{P}oincaré series},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {337},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {591--612},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11F30 (11F37)},
      MRNUMBER = {2274544},
      MRREVIEWER = {Paul M. Jenkins},
      DOI = {10.1007/s00208-006-0048-0},
      ZBLNUMBER = {1154.11015},
      }
  • [Br] Go to document J. H. Bruinier, "Borcherds products and Chern classes of Hirzebruch-Zagier divisors," Invent. Math., vol. 138, iss. 1, pp. 51-83, 1999.
    @article {Br, MRKEY = {1714336},
      AUTHOR = {Bruinier, Jan Hendrik},
      TITLE = {Borcherds products and {C}hern classes of {H}irzebruch-{Z}agier divisors},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {138},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {51--83},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F41 (11F55)},
      MRNUMBER = {1714336},
      MRREVIEWER = {Rainer Schulze-Pillot},
      DOI = {10.1007/s002220050341},
      ZBLNUMBER = {1011.11027},
      }
  • [BF1] Go to document J. H. Bruinier and J. Funke, "Traces of CM values of modular functions," J. Reine Angew. Math., vol. 594, pp. 1-33, 2006.
    @article {BF1, MRKEY = {2248151},
      AUTHOR = {Bruinier, Jan Hendrik and Funke, Jens},
      TITLE = {Traces of {CM} values of modular functions},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {594},
      YEAR = {2006},
      PAGES = {1--33},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11F37 (11F11)},
      MRNUMBER = {2248151},
      MRREVIEWER = {Rolf Berndt},
      DOI = {10.1515/CRELLE.2006.034},
      ZBLNUMBER = {1104.11021},
      }
  • [BF2] Go to document J. H. Bruinier and J. Funke, "On two geometric theta lifts," Duke Math. J., vol. 125, iss. 1, pp. 45-90, 2004.
    @article {BF2, MRKEY = {2097357},
      AUTHOR = {Bruinier, Jan Hendrik and Funke, Jens},
      TITLE = {On two geometric theta lifts},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {125},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {45--90},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11F55 (11F27 11F32)},
      MRNUMBER = {2097357},
      MRREVIEWER = {Matthew G. Boylan},
      DOI = {10.1215/S0012-7094-04-12513-8},
      ZBLNUMBER = {1088.11030},
      }
  • [CZ] Y. Choie and D. Zagier, "Rational period functions for ${ PSL}(2,\Bbb Z)$," in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Providence, RI: Amer. Math. Soc., 1993, vol. 143, pp. 89-108.
    @incollection {CZ, MRKEY = {1210514},
      AUTHOR = {Choie, Yj. and Zagier, D.},
      TITLE = {Rational period functions for {${\rm PSL}(2,\bold Z)$}},
      BOOKTITLE = {A Tribute to {E}mil {G}rosswald: Number Theory and Related Analysis},
      SERIES = {Contemp. Math.},
      VOLUME = {143},
      PAGES = {89--108},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1993},
      MRCLASS = {11F67 (11F25)},
      MRNUMBER = {1210514},
      MRREVIEWER = {Alexey A. Panchishkin},
      ZBLNUMBER = {0790.11044},
      }
  • [Du] Go to document W. Duke, "Modular functions and the uniform distribution of CM points," Math. Ann., vol. 334, iss. 2, pp. 241-252, 2006.
    @article {Du, MRKEY = {2207698},
      AUTHOR = {Duke, W.},
      TITLE = {Modular functions and the uniform distribution of {CM} points},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {334},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {241--252},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11F37},
      MRNUMBER = {2207698},
      MRREVIEWER = {Fran{ç}ois Martin},
      DOI = {10.1007/s00208-005-0706-7},
      ZBLNUMBER = {1091.11012},
      }
  • [DFI] Go to document W. Duke, J. B. Friedlander, and H. Iwaniec, "Equidistribution of roots of a quadratic congruence to prime moduli," Ann. of Math., vol. 141, iss. 2, pp. 423-441, 1995.
    @article {DFI, MRKEY = {1324141},
      AUTHOR = {Duke, W. and Friedlander, J. B. and Iwaniec, H.},
      TITLE = {Equidistribution of roots of a quadratic congruence to prime moduli},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {141},
      YEAR = {1995},
      NUMBER = {2},
      PAGES = {423--441},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11N64 (11L20)},
      MRNUMBER = {1324141},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.2307/2118527},
      ZBLNUMBER = {0840.11003},
      }
  • [DI] Go to document W. Duke and Ö. Imamo=glu, "A converse theorem and the Saito-Kurokawa lift," Internat. Math. Res. Notices, iss. 7, pp. 347-355, 1996.
    @article {DI, MRKEY = {1389957},
      AUTHOR = {Duke, W. and Imamo{\=g}lu, {Ö}.},
      TITLE = {A converse theorem and the {S}aito-{K}urokawa lift},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {1996},
      NUMBER = {7},
      PAGES = {347--355},
      ISSN = {1073-7928},
      MRCLASS = {11F32 (11F30 11F37 11F46 11F72)},
      MRNUMBER = {1389957},
      MRREVIEWER = {B. Ramakrishnan},
      DOI = {10.1155/S1073792896000220},
      ZBLNUMBER = {0849.11039},
      }
  • [DJ] Go to document W. Duke and P. Jenkins, "Integral traces of singular values of weak Maass forms," Algebra Number Theory, vol. 2, iss. 5, pp. 573-593, 2008.
    @article {DJ, MRKEY = {2429454},
      AUTHOR = {Duke, William and Jenkins, Paul},
      TITLE = {Integral traces of singular values of weak {M}aass forms},
      JOURNAL = {Algebra Number Theory},
      FJOURNAL = {Algebra \& Number Theory},
      VOLUME = {2},
      YEAR = {2008},
      NUMBER = {5},
      PAGES = {573--593},
      ISSN = {1937-0652},
      MRCLASS = {11F30 (11F37)},
      MRNUMBER = {2429454},
      MRREVIEWER = {Andre Reznikov},
      DOI = {10.2140/ant.2008.2.573},
      ZBLNUMBER = {05529338},
      }
  • [Ei] M. Eichler, "Grenzkreisgruppen und kettenbruchartige Algorithmen," Acta Arith., vol. 11, pp. 169-180, 1965.
    @article {Ei, MRKEY = {0228436},
      AUTHOR = {Eichler, M.},
      TITLE = {Grenzkreisgruppen und kettenbruchartige {A}lgorithmen},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      VOLUME = {11},
      YEAR = {1965},
      PAGES = {169--180},
      ISSN = {0065-1036},
      MRCLASS = {10.21},
      MRNUMBER = {0228436},
      MRREVIEWER = {J. Lehner},
      ZBLNUMBER = {0148.32503},
      }
  • [El] Go to document J. Elstrodt, "Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I," Math. Ann., vol. 203, pp. 295-300, 1973.
    @article {El, MRKEY = {0360472},
      AUTHOR = {Elstrodt, J{ü}rgen},
      TITLE = {Die {R}esolvente zum {E}igenwertproblem der automorphen {F}ormen in der hyperbolischen {E}bene. {I}},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {203},
      YEAR = {1973},
      PAGES = {295--300},
      ISSN = {0025-5831},
      MRCLASS = {10D15 (30A58)},
      MRNUMBER = {0360472},
      MRREVIEWER = {Rolf-Dieter Kulle},
      DOI = {10.1007/BF01351910},
      ZBLNUMBER = {0239.10014},
      }
  • [Fa] Go to document J. D. Fay, "Fourier coefficients of the resolvent for a Fuchsian group," J. Reine Angew. Math., vol. 293/294, pp. 143-203, 1977.
    @article {Fa, MRKEY = {0506038},
      AUTHOR = {Fay, John D.},
      TITLE = {Fourier coefficients of the resolvent for a {F}uchsian group},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
      VOLUME = {293/294},
      YEAR = {1977},
      PAGES = {143--203},
      ISSN = {0075-4102},
      MRCLASS = {10D15},
      MRNUMBER = {0506038},
      MRREVIEWER = {Marvin I. Knopp},
      DOI = {10.1515/crll.1977.293-294.143},
      ZBLNUMBER = {0352.30012},
      }
  • [F] Go to document J. Funke, "Heegner divisors and nonholomorphic modular forms," Compositio Math., vol. 133, iss. 3, pp. 289-321, 2002.
    @article {F, MRKEY = {1930980},
      AUTHOR = {Funke, Jens},
      TITLE = {Heegner divisors and nonholomorphic modular forms},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {133},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {289--321},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {11F37 (11F27 11F30)},
      MRNUMBER = {1930980},
      MRREVIEWER = {Jan Hendrik Bruinier},
      DOI = {10.1023/A:1020002121978},
      ZBLNUMBER = {1010.11023},
      }
  • [GH] Go to document D. Goldfeld and J. Hoffstein, "Eisenstein series of ${1\over 2}$-integral weight and the mean value of real Dirichlet $L$-series," Invent. Math., vol. 80, iss. 2, pp. 185-208, 1985.
    @article {GH, MRKEY = {0788407},
      AUTHOR = {Goldfeld, Dorian and Hoffstein, Jeffrey},
      TITLE = {Eisenstein series of {${1\over 2}$}-integral weight and the mean value of real {D}irichlet {$L$}-series},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {80},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {185--208},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {11F37 (11F30 11M41)},
      MRNUMBER = {0788407},
      MRREVIEWER = {L. A. Takhtajan (Takhtadzhyan)},
      DOI = {10.1007/BF01388603},
      ZBLNUMBER = {0564.10043},
      }
  • [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Fifth ed., Boston, MA: Academic Press, 1994.
    @book {GR, MRKEY = {1243179},
      AUTHOR = {Gradshteyn, I. S. and Ryzhik, I. M.},
      TITLE = {Table of Integrals, Series, and Products},
      EDITION = {{F}ifth},
      PUBLISHER = {Academic Press},
      ADDRESS = {Boston, MA},
      YEAR = {1994},
      PAGES = {xlviii+1204},
      ISBN = {0-12-294755-X},
      MRCLASS = {00A22 (33-00 65-00)},
      MRNUMBER = {1243179},
      ZBLNUMBER = {0918.65002},
      }
  • [GKZ] Go to document B. Gross, W. Kohnen, and D. Zagier, "Heegner points and derivatives of $L$-series. II," Math. Ann., vol. 278, iss. 1-4, pp. 497-562, 1987.
    @article {GKZ, MRKEY = {0909238},
      AUTHOR = {Gross, B. and Kohnen, W. and Zagier, D.},
      TITLE = {Heegner points and derivatives of {$L$}-series. {II}},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {278},
      YEAR = {1987},
      NUMBER = {1-4},
      PAGES = {497--562},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11G40 (11F30 11G05 11R29 14G10)},
      MRNUMBER = {0909238},
      MRREVIEWER = {Lawrence Washington},
      DOI = {10.1007/BF01458081},
      ZBLNUMBER = {0079.03302},
      }
  • [HZ] Go to document F. Hirzebruch and D. Zagier, "Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus," Invent. Math., vol. 36, pp. 57-113, 1976.
    @article {HZ, MRKEY = {0453649},
      AUTHOR = {Hirzebruch, F. and Zagier, D.},
      TITLE = {Intersection numbers of curves on {H}ilbert modular surfaces and modular forms of {N}ebentypus},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {36},
      YEAR = {1976},
      PAGES = {57--113},
      ISSN = {0020-9910},
      MRCLASS = {10D20 (14J25)},
      MRNUMBER = {0453649},
      MRREVIEWER = {Hiroshi Saito},
      DOI = {10.1007/BF01390005},
      ZBLNUMBER = {0332.14009},
      }
  • [Ho0] Go to document C. Hooley, "An asymptotic formula in the theory of numbers," Proc. London Math. Soc., vol. 7, pp. 396-413, 1957.
    @article {Ho0, MRKEY = {0090613},
      AUTHOR = {Hooley, Christopher},
      TITLE = {An asymptotic formula in the theory of numbers},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {7},
      YEAR = {1957},
      PAGES = {396--413},
      ISSN = {0024-6115},
      MRCLASS = {10.1X},
      MRNUMBER = {0090613},
      MRREVIEWER = {A. L. Whiteman},
      DOI = {10.1112/plms/s3-7.1.396},
      ZBLNUMBER = {0079.27301},
      }
  • [Ho] Go to document C. Hooley, "On the number of divisors of a quadratic polynomial," Acta Math., vol. 110, pp. 97-114, 1963.
    @article {Ho, MRKEY = {0153648},
      AUTHOR = {Hooley, Christopher},
      TITLE = {On the number of divisors of a quadratic polynomial},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {110},
      YEAR = {1963},
      PAGES = {97--114},
      ISSN = {0001-5962},
      MRCLASS = {10.45 (10.43)},
      MRNUMBER = {0153648},
      MRREVIEWER = {P. Erd{ö}s},
      DOI = {10.1007/BF02391856},
      ZBLNUMBER = {0116.03802},
      }
  • [IS] T. Ibukiyama and H. Saito, On zeta functions associated to symmetric matrices (II): Functional equations and special values.
    @misc{IS,
      author={Ibukiyama, T. and Saito, H.},
      TITLE={On zeta functions associated to symmetric matrices ({II}): {F}unctional equations and special values},
      NOTE={{MPI} preprint 1997-37; \href{http://www.mpim-bonn.mpg.de/node/263}{http://www.mpim-bonn.mpg.de/node/263}},
      }
  • [IK] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
    @book {IK, MRKEY = {2061214},
      AUTHOR = {Iwaniec, Henryk and Kowalski, Emmanuel},
      TITLE = {Analytic Number Theory},
      SERIES = {Amer. Math. Soc. Colloq. Publ.},
      VOLUME = {53},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      PAGES = {xii+615},
      ISBN = {0-8218-3633-1},
      MRCLASS = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
      MRNUMBER = {2061214},
      MRREVIEWER = {K. Soundararajan},
      ZBLNUMBER = {0641.14013},
      }
  • [Je] Go to document P. Jenkins, "Kloosterman sums and traces of singular moduli," J. Number Theory, vol. 117, iss. 2, pp. 301-314, 2006.
    @article {Je, MRKEY = {2213767},
      AUTHOR = {Jenkins, Paul},
      TITLE = {Kloosterman sums and traces of singular moduli},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {117},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {301--314},
      ISSN = {0022-314X},
      CODEN = {JNUTA9},
      MRCLASS = {11F37},
      MRNUMBER = {2213767},
      MRREVIEWER = {Matthew G. Boylan},
      DOI = {10.1016/j.jnt.2005.06.009},
      ZBLNUMBER = {1177.11038},
      }
  • [kaneko] Go to document M. Kaneko, "Observations on the `values’ of the elliptic modular function $j(\tau)$ at real quadratics," Kyushu J. Math., vol. 63, iss. 2, pp. 353-364, 2009.
    @article {kaneko, MRKEY = {2568779},
      AUTHOR = {Kaneko, Masanobu},
      TITLE = {Observations on the `values' of the elliptic modular function {$j(\tau)$} at real quadratics},
      JOURNAL = {Kyushu J. Math.},
      FJOURNAL = {Kyushu Journal of Mathematics},
      VOLUME = {63},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {353--364},
      ISSN = {1340-6116},
      MRCLASS = {11F30 (11F03 11Y70)},
      MRNUMBER = {2568779},
      MRREVIEWER = {David Y. Jao},
      DOI = {10.2206/kyushujm.63.353},
      ZBLNUMBER = {pre05640294},
      }
  • [KS] Go to document S. Katok and P. Sarnak, "Heegner points, cycles and Maass forms," Israel J. Math., vol. 84, iss. 1-2, pp. 193-227, 1993.
    @article {KS, MRKEY = {1244668},
      AUTHOR = {Katok, Svetlana and Sarnak, Peter},
      TITLE = {Heegner points, cycles and {M}aass forms},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {84},
      YEAR = {1993},
      NUMBER = {1-2},
      PAGES = {193--227},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {11F67 (11F27 11F32 11F37)},
      MRNUMBER = {1244668},
      MRREVIEWER = {George Gilbert},
      DOI = {10.1007/BF02761700},
      ZBLNUMBER = {0787.11016},
      }
  • [Kn1] Go to document M. I. Knopp, "Some new results on the Eichler cohomology of automorphic forms," Bull. Amer. Math. Soc., vol. 80, pp. 607-632, 1974.
    @article {Kn1, MRKEY = {0344454},
      AUTHOR = {Knopp, Marvin I.},
      TITLE = {Some new results on the {E}ichler cohomology of automorphic forms},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {Bulletin of the American Mathematical Society},
      VOLUME = {80},
      YEAR = {1974},
      PAGES = {607--632},
      ISSN = {0002-9904},
      MRCLASS = {30A58 (10D15 32N10)},
      MRNUMBER = {0344454},
      MRREVIEWER = {I. Kra},
      DOI = {10.1090/S0002-9904-1974-13520-2},
      ZBLNUMBER = {1059.11001},
      }
  • [Kn2] Go to document M. I. Knopp, "Rational period functions of the modular group," Duke Math. J., vol. 45, iss. 1, pp. 47-62, 1978.
    @article {Kn2, MRKEY = {0485700},
      AUTHOR = {Knopp, Marvin I.},
      TITLE = {Rational period functions of the modular group},
      NOTE = {With an appendix by {G}eorges {G}rinstein},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {45},
      YEAR = {1978},
      NUMBER = {1},
      PAGES = {47--62},
      ISSN = {0012-7094},
      MRCLASS = {10D10},
      MRNUMBER = {0485700},
      MRREVIEWER = {J. Lehner},
      URL = {http://projecteuclid.org/getRecord?id=euclid.dmj/1077312686},
      ZBLNUMBER = {0374.10014},
      }
  • [Ko1] W. Kohnen, Beziehungen Zwischen Modulformen Halbganzen Gewichts und Modulformen Ganzen Gewichts, Bonn: Universität Bonn Mathematisches Institut, 1981, vol. 131.
    @book {Ko1, MRKEY = {0633060},
      AUTHOR = {Kohnen, Winfried},
      TITLE = {Beziehungen Zwischen {M}odulformen Halbganzen {G}ewichts und {M}odulformen Ganzen {G}ewichts},
      SERIES = {Bonner Mathematische Schriften [Bonn Math. Publ.]},
      VOLUME={131},
      NOTE = {Dissertation, Rheinische Friedrich-Wilhelms-Universit{ä}t, Bonn, 1980},
      PUBLISHER = {Universität Bonn Mathematisches Institut},
      ADDRESS = {Bonn},
      YEAR = {1981},
      PAGES = {v+104},
      MRCLASS = {10D12},
      MRNUMBER = {0633060},
      MRREVIEWER = {K.-B. Gundlach},
      ZBLNUMBER = {0451.10016},
      }
  • [Ko2] Go to document W. Kohnen, "Modular forms of half-integral weight on $\Gamma _{0}(4)$," Math. Ann., vol. 248, iss. 3, pp. 249-266, 1980.
    @article {Ko2, MRKEY = {0575942},
      AUTHOR = {Kohnen, Winfried},
      TITLE = {Modular forms of half-integral weight on {$\Gamma \sb{0}(4)$}},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {248},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {249--266},
      ISSN = {0025-5831},
      CODEN = {MAANA3},
      MRCLASS = {10D12},
      MRNUMBER = {0575942},
      MRREVIEWER = {Don Zagier},
      DOI = {10.1007/BF01420529},
      ZBLNUMBER = {0416.10023},
      }
  • [Ko] Go to document W. Kohnen, "Fourier coefficients of modular forms of half-integral weight," Math. Ann., vol. 271, iss. 2, pp. 237-268, 1985.
    @article {Ko, MRKEY = {0783554},
      AUTHOR = {Kohnen, Winfried},
      TITLE = {Fourier coefficients of modular forms of half-integral weight},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {271},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {237--268},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11F11},
      MRNUMBER = {0783554},
      MRREVIEWER = {Toyokazu Hiramatsu},
      DOI = {10.1007/BF01455989},
      ZBLNUMBER = {0542.10018},
      }
  • [KZ] Go to document W. Kohnen and D. Zagier, "Values of $L$-series of modular forms at the center of the critical strip," Invent. Math., vol. 64, iss. 2, pp. 175-198, 1981.
    @article {KZ, MRKEY = {0629468},
      AUTHOR = {Kohnen, Winfried and Zagier, D.},
      TITLE = {Values of {$L$}-series of modular forms at the center of the critical strip},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {64},
      YEAR = {1981},
      NUMBER = {2},
      PAGES = {175--198},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {10D24 (10D12)},
      MRNUMBER = {0629468},
      MRREVIEWER = {Marie-France Vign{é}ras},
      DOI = {10.1007/BF01389166},
      ZBLNUMBER = {0468.10015},
      }
  • [Le] N. N. Lebedev, Special Functions and Their Applications, New York: Dover Publications, 1972.
    @book {Le, MRKEY = {0350075},
      AUTHOR = {Lebedev, N. N.},
      TITLE = {Special Functions and Their Applications},
      NOTE = {revised edition, translated from the {R}ussian and edited by {R}ichard {A}. {S}ilverman; unabridged and corrected republication},
      PUBLISHER = {Dover Publications},
      ADDRESS = {New York},
      YEAR = {1972},
      PAGES = {xii+308},
      MRCLASS = {33-XX (69.00)},
      MRNUMBER = {0350075},
      ZBLNUMBER = {0271.33001},
      }
  • [MO] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, New York: Springer-Verlag, 1966, vol. 52.
    @book {MO, MRKEY = {0232968},
      AUTHOR = {Magnus, Wilhelm and Oberhettinger, Fritz and Soni, Raj Pal},
      TITLE = {Formulas and Theorems for the Special Functions of Mathematical Physics},
      SERIES = {Third enlarged edition. Grundlehren Math. Wissen.},
      VOLUME={52},
      PUBLISHER = {Springer-Verlag},
      YEAR = {1966},
      PAGES = {viii+508},
      MRCLASS = {33.00},
      MRNUMBER = {0232968},
      MRREVIEWER = {Mary L. Boas},
      ADDRESS = {New York},
      ZBLNUMBER = {0143.08502},
      }
  • [Ne] H. Neunhöffer, "Über die analytische Fortsetzung von Poincaréreihen," S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl., pp. 33-90, 1973.
    @article {Ne, MRKEY = {0352007},
      AUTHOR = {Neunh{ö}ffer, H.},
      TITLE = {Über die analytische {F}ortsetzung von {P}oincaréreihen},
      JOURNAL = {S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl.},
      YEAR = {1973},
      PAGES = {33--90},
      MRCLASS = {10D15 (10D10 30A58)},
      MRNUMBER = {0352007},
      MRREVIEWER = {Jurgen Elstrodt},
      ZBLNUMBER = {0272.10015},
      }
  • [Ni] Go to document D. Niebur, "A class of nonanalytic automorphic functions," Nagoya Math. J., vol. 52, pp. 133-145, 1973.
    @article {Ni, MRKEY = {0337788},
      AUTHOR = {Niebur, Douglas},
      TITLE = {A class of nonanalytic automorphic functions},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {52},
      YEAR = {1973},
      PAGES = {133--145},
      ISSN = {0027-7630},
      MRCLASS = {10D10 (10D15 30A58)},
      MRNUMBER = {0337788},
      MRREVIEWER = {Jurgen Elstrodt},
      URL = {http://projecteuclid.org/getRecord?id=euclid.nmj/1118794882},
      ZBLNUMBER = {0288.10010},
      }
  • [On] K. Ono, "Unearthing the visions of a master: harmonic Maass forms and number theory," in Current Developments in Mathematics, 2008, Somerville, MA: Internat. Press, 2009, pp. 347-454.
    @incollection {On, MRKEY = {2555930},
      AUTHOR = {Ono, Ken},
      TITLE = {Unearthing the visions of a master: harmonic {M}aass forms and number theory},
      BOOKTITLE = {Current Developments in Mathematics, 2008},
      PAGES = {347--454},
      PUBLISHER = {Internat. Press},
      ADDRESS={ Somerville, MA},
      YEAR = {2009},
      MRCLASS = {11F37 (11-02 11F11 11P82 11P84)},
      MRNUMBER = {2555930},
      MRREVIEWER = {Kathrin Bringmann},
      ZBLNUMBER = {05635202},
      }
  • [Pa] A. L. Parson, "Modular integrals and indefinite binary quadratic forms," in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Providence, RI: Amer. Math. Soc., 1993, vol. 143, pp. 513-523.
    @incollection {Pa, MRKEY = {1210538},
      AUTHOR = {Parson, L. Alayne},
      TITLE = {Modular integrals and indefinite binary quadratic forms},
      BOOKTITLE = {A Tribute to {E}mil {G}rosswald: Number Theory and Related Analysis},
      SERIES = {Contemp. Math.},
      VOLUME = {143},
      PAGES = {513--523},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1993},
      MRCLASS = {11F67 (11E45 11F25)},
      MRNUMBER = {1210538},
      MRREVIEWER = {Rainer Schulze-Pillot},
      ZBLNUMBER = {0795.11022},
      }
  • [Ro] Go to document W. Roelcke, "Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II. (German)," Math. Ann., vol. 167, pp. 292-337, 1966.
    @article{Ro,
      author={Roelcke, W.},
      TITLE={Das {E}igenwertproblem der automorphen {F}ormen in der hyperbolischen {E}bene. {I, II}. ({G}erman)},
      JOURNAL={Math. Ann.},
      VOLUME={167},
      YEAR={1966},
      PAGES={292--337},
      NOTE={{\it ibid}. {\bf 168} (1966), 261--324},
      MRNUMBER = {0243062},
      ZBLNUMBER = {0152.07705},
      DOI = {10.1007/BF01364540},
      DOI = {10.1007/BF01361556},
      }
  • [Sh] Go to document G. Shimura, "On modular forms of half integral weight," Ann. of Math., vol. 97, pp. 440-481, 1973.
    @article {Sh, MRKEY = {0332663},
      AUTHOR = {Shimura, Goro},
      TITLE = {On modular forms of half integral weight},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {97},
      YEAR = {1973},
      PAGES = {440--481},
      ISSN = {0003-486X},
      MRCLASS = {10D15},
      MRNUMBER = {0332663},
      MRREVIEWER = {I. I. Pjateckii-Sapiro},
      DOI = {10.2307/1970831},
      ZBLNUMBER = {0266.10022},
      }
  • [To] Go to document &. Tóth, "On the evaluation of Salié sums," Proc. Amer. Math. Soc., vol. 133, iss. 3, pp. 643-645, 2005.
    @article {To, MRKEY = {2113909},
      AUTHOR = {T{ó}th, {Á}rp{á}d},
      TITLE = {On the evaluation of {S}alié sums},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {133},
      YEAR = {2005},
      NUMBER = {3},
      PAGES = {643--645},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {11L05 (11F37)},
      MRNUMBER = {2113909},
      MRREVIEWER = {Ronald J. Evans},
      DOI = {10.1090/S0002-9939-04-07768-8},
      ZBLNUMBER = {1092.11036},
      }
  • [Wa] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge, England: Cambridge Univ. Press, 1944.
    @book {Wa, MRKEY = {0010746},
      AUTHOR = {Watson, G. N.},
      TITLE = {A {T}reatise on the {T}heory of {B}essel {F}unctions},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge, England},
      YEAR = {1944},
      PAGES = {vi+804},
      MRCLASS = {33.0X},
      MRNUMBER = {0010746},
      MRREVIEWER = {G. Szeg{ö}},
      ZBLNUMBER = {0063.08184},
      }
  • [We] A. Weil, "On some exponential sums," Proc. Nat. Acad. Sci. U. S. A., vol. 34, pp. 204-207, 1948.
    @article {We, MRKEY = {0027006},
      AUTHOR = {Weil, Andr{é}},
      TITLE = {On some exponential sums},
      JOURNAL = {Proc. Nat. Acad. Sci. U. S. A.},
      FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
      VOLUME = {34},
      YEAR = {1948},
      PAGES = {204--207},
      ISSN = {0027-8424},
      MRCLASS = {10.0X},
      MRNUMBER = {0027006},
      MRREVIEWER = {H. D. Kloosterman},
      ZBLNUMBER = {0032.26102},
      }
  • [WW] E. T. Whittaker and G. N. Watson, 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, Fourth; reprinted ed., New York: Cambridge Univ. Press, 1962.
    @book {WW, MRKEY = {0178117},
      AUTHOR = {Whittaker, E. T. and Watson, G. N.},
      TITLE = {A Course of Modern Analysis. {A}n Introduction to the General Theory of Infinite Processes and of Analytic Functions: With an Account of the Principal Transcendental Functions},
      EDITION = {{F}ourth; reprinted},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {New York},
      YEAR = {1962},
      PAGES = {vii+608},
      MRCLASS = {30.00},
      MRNUMBER = {0178117},
      ZBLNUMBER = {0105.26901},
      }
  • [Z] Go to document D. Zagier, "Modular forms associated to real quadratic fields," Invent. Math., vol. 30, iss. 1, pp. 1-46, 1975.
    @article {Z, MRKEY = {0382174},
      AUTHOR = {Zagier, Don},
      TITLE = {Modular forms associated to real quadratic fields},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {30},
      YEAR = {1975},
      NUMBER = {1},
      PAGES = {1--46},
      ISSN = {0020-9910},
      MRCLASS = {10D05 (10D20)},
      MRNUMBER = {0382174},
      MRREVIEWER = {Larry J. Goldstein},
      DOI = {10.1007/BF01389846},
      ZBLNUMBER = {0308.10014},
      }
  • [Z1] D. Zagier, "Nombres de classes et formes modulaires de poids $3/2$," C. R. Acad. Sci. Paris Sér. A, vol. 281, iss. 21, pp. 883-886, 1975.
    @article {Z1, MRKEY = {0429750},
      AUTHOR = {Zagier, Don},
      TITLE = {Nombres de classes et formes modulaires de poids {$3/2$}},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. A},
      VOLUME = {281},
      YEAR = {1975},
      NUMBER = {21},
      PAGES = {883--886},
      MRCLASS = {10D15 (12A50 14J25)},
      MRNUMBER = {0429750},
      MRREVIEWER = {William F. Hammond},
      ZBLNUMBER = {0323.10021},
     }
  • [Za3] D. Zagier, "Eisenstein series and the Riemann zeta function," in Automorphic Forms, Representation Theory and Arithmetic, 1981, pp. 275-301.
    @inproceedings{Za3, MRKEY = {0633666},
      AUTHOR = {Zagier, Don},
      TITLE = {Eisenstein series and the {R}iemann zeta function},
      BOOKTITLE = {Automorphic Forms, Representation Theory and Arithmetic},
      VENUE={{B}ombay, 1979},
      SERIES = {Tata Inst. Fund. Res. Studies in Math.},
      VOLUME = {10},
      PAGES = {275--301},
      PUBLISHER = {Tata Inst. Fundamental Res., Bombay},
      YEAR = {1981},
      MRCLASS = {10D15 (10D24 10D40 10H05)},
      MRNUMBER = {0633666},
      MRREVIEWER = {Audrey A. Terras},
      ZBLNUMBER = {0484.10019},
      }
  • [Za] D. Zagier, "Traces of singular moduli," in Motives, Polylogarithms and Hodge Theory, Part I, Somerville, MA: Internat. Press, 2002, vol. 3, pp. 211-244.
    @incollection {Za, MRKEY = {1977587},
      AUTHOR = {Zagier, Don},
      TITLE = {Traces of singular moduli},
      BOOKTITLE = {Motives, Polylogarithms and {H}odge Theory, {P}art {\rm I}},
      VENUE={{I}rvine, {CA},
      1998},
      SERIES = {Internat. Press Lect. Ser.},
      VOLUME= {3},
      PAGES = {211--244},
      PUBLISHER = {Internat. Press},
      ADDRESS={Somerville, MA},
      YEAR = {2002},
      MRCLASS = {11F11 (11F03 11F30 11F37)},
      MRNUMBER = {1977587},
      MRREVIEWER = {Rainer Schulze-Pillot},
      ZBLNUMBER = {1048.11035},
      }
  • [Za2] D. Zagier, "Ramanujan’s mock theta functions and their applications [d’après Zwegers and Bringmann-Ono]," Sem. Bourbaki, vol. 326, pp. 143-164, 2009.
    @article{Za2,
      author = {Zagier, Don},
      TITLE = {{R}amanujan's mock theta functions and their applications [d'après {Z}wegers and {B}ringmann-{O}no]},
      JOURNAL={Sem. Bourbaki},
      VOLUME={326},
      YEAR={2009},
      PAGES={143--164},
      NOTE={Exp. {N}o. 986},
      MRNUMBER = {2605321},
      ZBLNUMBER = {1198.11046},
      }
  • [Zw1] Go to document S. P. Zwegers, Mock theta functions, 2002.
    @misc{Zw1,
      author={Zwegers, S. P.},
      TITLE={Mock theta functions},
      NOTE={{U}trecht {P}h.{D}. thesis},
      YEAR={2002},
      URL={http://igitur-archive.library.uu.nl/dissertations/2003-0127-094324/inhoud.htm},
      }
  • [Zw2] S. P. Zwegers, "Mock $\theta$-functions and real analytic modular forms," in $q$-Series with Applications to Combinatorics, Number Theory, and Physics, Providence, RI: Amer. Math. Soc., 2001, vol. 291, pp. 269-277.
    @incollection {Zw2, MRKEY = {1874536},
      AUTHOR = {Zwegers, S. P.},
      TITLE = {Mock {$\theta$}-functions and real analytic modular forms},
      BOOKTITLE = {{$q$}-Series with Applications to Combinatorics, Number Theory, and Physics},
      VENUE={{U}rbana, {IL},
      2000},
      SERIES = {Contemp. Math.},
      VOLUME= {291},
      PAGES = {269--277},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2001},
      MRCLASS = {11F27 (33D15)},
      MRNUMBER = {1874536},
      MRREVIEWER = {Bruce C. Berndt},
      ZBLNUMBER = {1044.11029},
      }

Authors

William Duke

University of California
Los Angeles, CA

Özlem Imamoglu

ETH
Zürich
Switzerland

Á. Tóth

Eotvos Lorand University
Department of Analysis
Budapest
Hungary