×

zbMATH — the first resource for mathematics

Cycle integrals of the \(j\)-function and mock modular forms. (English) Zbl 1270.11044
Summary: 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.

MSC:
11F30 Fourier coefficients of automorphic forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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. · Zbl 0907.11013
[2] R. E. Borcherds, ”Automorphic forms on \({ O}_{s+2,2}({\mathbf R})\) and infinite products,” Invent. Math., vol. 120, iss. 1, pp. 161-213, 1995. · Zbl 0932.11028 · doi:10.1007/BF01241126 · eudml:144273
[3] 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. · Zbl 1154.11015 · doi:10.1007/s00208-006-0048-0
[4] J. H. Bruinier, ”Borcherds products and Chern classes of Hirzebruch-Zagier divisors,” Invent. Math., vol. 138, iss. 1, pp. 51-83, 1999. · Zbl 1011.11027 · doi:10.1007/s002220050341
[5] J. H. Bruinier and J. Funke, ”Traces of CM values of modular functions,” J. Reine Angew. Math., vol. 594, pp. 1-33, 2006. · Zbl 1104.11021 · doi:10.1515/CRELLE.2006.034 · arxiv:math/0408406
[6] J. H. Bruinier and J. Funke, ”On two geometric theta lifts,” Duke Math. J., vol. 125, iss. 1, pp. 45-90, 2004. · Zbl 1088.11030 · doi:10.1215/S0012-7094-04-12513-8 · arxiv:math/0212286
[7] Y. Choie and D. Zagier, ”Rational period functions for \({ PSL}(2,\mathbb Z)\),” in A Tribute to Emil Grosswald: Number Theory and Related Analysis, Providence, RI: Amer. Math. Soc., 1993, vol. 143, pp. 89-108. · Zbl 0790.11044
[8] W. Duke, ”Modular functions and the uniform distribution of CM points,” Math. Ann., vol. 334, iss. 2, pp. 241-252, 2006. · Zbl 1091.11012 · doi:10.1007/s00208-005-0706-7
[9] 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. · Zbl 0840.11003 · doi:10.2307/2118527
[10] W. Duke and Ö. Imamo=glu, ”A converse theorem and the Saito-Kurokawa lift,” Internat. Math. Res. Notices, iss. 7, pp. 347-355, 1996. · Zbl 0849.11039 · doi:10.1155/S1073792896000220
[11] 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. · Zbl 1215.11046 · doi:10.2140/ant.2008.2.573 · pjm.math.berkeley.edu
[12] M. Eichler, ”Grenzkreisgruppen und kettenbruchartige Algorithmen,” Acta Arith., vol. 11, pp. 169-180, 1965. · Zbl 0148.32503 · eudml:204752
[13] J. Elstrodt, ”Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I,” Math. Ann., vol. 203, pp. 295-300, 1973. · Zbl 0239.10014 · doi:10.1007/BF01351910 · eudml:162450
[14] J. D. Fay, ”Fourier coefficients of the resolvent for a Fuchsian group,” J. Reine Angew. Math., vol. 293/294, pp. 143-203, 1977. · Zbl 0352.30012 · doi:10.1515/crll.1977.293-294.143 · crelle:GDZPPN002193825 · eudml:151896
[15] J. Funke, ”Heegner divisors and nonholomorphic modular forms,” Compositio Math., vol. 133, iss. 3, pp. 289-321, 2002. · Zbl 1010.11023 · doi:10.1023/A:1020002121978
[16] 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. · Zbl 0564.10043 · doi:10.1007/BF01388603 · eudml:143227
[17] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Fifth ed., Boston, MA: Academic Press, 1994. · Zbl 0918.65002
[18] 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. · Zbl 0079.03302 · doi:10.2307/1992959
[19] 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. · Zbl 0332.14009 · doi:10.1007/BF01390005 · eudml:142414
[20] C. Hooley, ”An asymptotic formula in the theory of numbers,” Proc. London Math. Soc., vol. 7, pp. 396-413, 1957. · Zbl 0079.27301 · doi:10.1112/plms/s3-7.1.396
[21] C. Hooley, ”On the number of divisors of a quadratic polynomial,” Acta Math., vol. 110, pp. 97-114, 1963. · Zbl 0116.03802 · doi:10.1007/BF02391856
[22] T. Ibukiyama and H. Saito, On zeta functions associated to symmetric matrices (II): Functional equations and special values. · Zbl 1275.11085 · doi:10.1215/00277630-1815258 · euclid:nmj/1354716562
[23] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53. · Zbl 0641.14013 · doi:10.1007/BF01458081 · eudml:164302
[24] P. Jenkins, ”Kloosterman sums and traces of singular moduli,” J. Number Theory, vol. 117, iss. 2, pp. 301-314, 2006. · Zbl 1177.11038 · doi:10.1016/j.jnt.2005.06.009 · math.byu.edu
[25] 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. · Zbl 1246.11096 · doi:10.2206/kyushujm.63.353 · arxiv:0905.3013
[26] S. Katok and P. Sarnak, ”Heegner points, cycles and Maass forms,” Israel J. Math., vol. 84, iss. 1-2, pp. 193-227, 1993. · Zbl 0787.11016 · doi:10.1007/BF02761700
[27] M. I. Knopp, ”Some new results on the Eichler cohomology of automorphic forms,” Bull. Amer. Math. Soc., vol. 80, pp. 607-632, 1974. · Zbl 1059.11001 · doi:10.1090/S0002-9904-1974-13520-2
[28] M. I. Knopp, ”Rational period functions of the modular group,” Duke Math. J., vol. 45, iss. 1, pp. 47-62, 1978. · Zbl 0374.10014 · doi:10.1215/S0012-7094-78-04504-0 · projecteuclid.org
[29] W. Kohnen, Beziehungen Zwischen Modulformen Halbganzen Gewichts und Modulformen Ganzen Gewichts, Bonn: Universität Bonn Mathematisches Institut, 1981, vol. 131. · Zbl 0451.10016
[30] W. Kohnen, ”Modular forms of half-integral weight on \(\Gamma _0(4)\),” Math. Ann., vol. 248, iss. 3, pp. 249-266, 1980. · Zbl 0416.10023 · doi:10.1007/BF01420529 · eudml:163382
[31] W. Kohnen, ”Fourier coefficients of modular forms of half-integral weight,” Math. Ann., vol. 271, iss. 2, pp. 237-268, 1985. · Zbl 0542.10018 · doi:10.1007/BF01455989 · eudml:163966
[32] 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. · Zbl 0468.10015 · doi:10.1007/BF01389166 · eudml:142819
[33] N. N. Lebedev, Special Functions and Their Applications, New York: Dover Publications, 1972. · Zbl 0271.33001
[34] 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. · Zbl 0143.08502
[35] H. Neunhöffer, ”Über die analytische Fortsetzung von Poincaréreihen,” S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl., pp. 33-90, 1973. · Zbl 0272.10015
[36] D. Niebur, ”A class of nonanalytic automorphic functions,” Nagoya Math. J., vol. 52, pp. 133-145, 1973. · Zbl 0288.10010 · projecteuclid.org
[37] 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. · Zbl 1229.11074 · euclid:cdm/1254748659
[38] 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. · Zbl 0795.11022
[39] W. Roelcke, ”Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II. (German),” Math. Ann., vol. 167, pp. 292-337, 1966. · Zbl 0152.07705 · doi:10.1007/BF01364540 · eudml:161488
[40] G. Shimura, ”On modular forms of half integral weight,” Ann. of Math., vol. 97, pp. 440-481, 1973. · Zbl 0266.10022 · doi:10.2307/1970831
[41] &. Tóth, ”On the evaluation of Salié sums,” Proc. Amer. Math. Soc., vol. 133, iss. 3, pp. 643-645, 2005. · Zbl 1092.11036 · doi:10.1090/S0002-9939-04-07768-8
[42] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge, England: Cambridge Univ. Press, 1944. · Zbl 0063.08184
[43] A. Weil, ”On some exponential sums,” Proc. Nat. Acad. Sci. U. S. A., vol. 34, pp. 204-207, 1948. · Zbl 0032.26102 · doi:10.1073/pnas.34.5.204
[44] 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. · Zbl 0105.26901
[45] D. Zagier, ”Modular forms associated to real quadratic fields,” Invent. Math., vol. 30, iss. 1, pp. 1-46, 1975. · Zbl 0308.10014 · doi:10.1007/BF01389846 · eudml:142343
[46] 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. · Zbl 0323.10021 · eudml:181964
[47] D. Zagier, ”Eisenstein series and the Riemann zeta function,” in Automorphic Forms, Representation Theory and Arithmetic, 1981, pp. 275-301. · Zbl 0484.10019
[48] D. Zagier, ”Traces of singular moduli,” in Motives, Polylogarithms and Hodge Theory, Part I, Somerville, MA: Internat. Press, 2002, vol. 3, pp. 211-244. · Zbl 1048.11035
[49] D. Zagier, ”Ramanujan’s mock theta functions and their applications [d’après Zwegers and Bringmann-Ono],” Sem. Bourbaki, vol. 326, pp. 143-164, 2009. · Zbl 1198.11046 · smf4.emath.fr
[50] S. P. Zwegers, Mock theta functions, 2002. · Zbl 1194.11058 · igitur-archive.library.uu.nl
[51] 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. · Zbl 1044.11029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.