# zbMATH — the first resource for mathematics

Regularized inner products of modular functions. (English) Zbl 1418.11069
Summary: In this note, we give an explicit basis for the harmonic weak forms of weight two. We also show that their holomorphic coefficients can be given in terms of regularized inner products of weight zero weakly holomorphic forms.

##### MSC:
 11F12 Automorphic forms, one variable 11F30 Fourier coefficients of automorphic forms 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Full Text:
##### References:
 [1] Borcherds, R.E., Automorphic forms with singularities on Grassmannians, Invent. Math., 132, 491-562, (1998) · Zbl 0919.11036 [2] Bruggeman, R.: Harmonic lifts of modular forms (preprint). arXiv:1206.5118v1 [3] Bruinier, J.H., Borcherds products and Chern classes of Hirzebruch-Zagier divisors, Invent. Math., 138, 51-83, (1999) · Zbl 1011.11027 [4] Bruinier, J.H.; Funke, J., On two geometric theta lifts, Duke Math. J., 125, 45-90, (2004) · Zbl 1088.11030 [5] Bucholz: The Confluent Hypergeometric Function with Special Emphasis on Its Applications. Springer, New York (1969). xviii+238 pp. [6] Duke, W.; Imamoglu, O.; Tóth, Á., Cycle integrals of the j-function and mock modular forms, Ann. Math., 173, 947-981, (2011) · Zbl 1270.11044 [7] Duke, W.; Imamoglu, O.; Tóth, Á., Rational period functions and cycle integrals, Abh. Math. Semin. Univ. Hamb., 80, 255-264, (2010) · Zbl 1269.11040 [8] Duke, W.; Jenkins, P., Integral traces of singular values of weak Maass forms, Algebra Number Theory, 2, 573-593, (2008) · Zbl 1215.11046 [9] Fay, J.D., Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math., 293/294, 143-203, (1977) · Zbl 0352.30012 [10] Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. Am. Math. Soc., Providence (2004) · Zbl 1059.11001 [11] Jeon, D.; Kang, S.-Y.; Kim, C.H., Weak Maass-Poincaré series and weight 3/2 mock modular forms, J. Number Theory, 133, 2567-2587, (2013) · Zbl 1287.11052 [12] Knopp, M.I., Rademacher on $$J$$($$τ$$), Poincaré series of nonpositive weights and the eichler cohomology, Not. Am. Math. Soc., 37, 385-393, (1990) · Zbl 0695.10021 [13] Miller, P.D.: Applied Asymptotic Analysis. Am. Math. Soc., Providence (2006) · Zbl 1101.41031 [14] Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics, third enlarged edn. Die Grundlehren der Mathematischen Wissenschaften, vol. 52. Springer, New York (1966) · Zbl 0143.08502 [15] Rademacher, H., The Fourier coefficients of the modular invariant $$J$$($$τ$$), Am. J. Math., 60, 501-512, (1938) · JFM 64.0122.01 [16] Ramanujan, S.: On Certain Trigonometrical Sums and Their Applications in the Theory of Numbers. Collected Papers of Srinivasa Ramanujan, vol. 179. AMS, Providence (2000) [17] Watson, G.N., The harmonic functions associated with the parabolic cylinder, Proc. Lond. Math. Soc., S2-8, 393, (1918) · JFM 41.0531.01
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.