Cycle integrals of a sesqui-harmonic Maass form of weight zero

JOURNAL OF NUMBER THEORY

卷:141

Article

Jeon, Daeyeol¹  Kang, Soon-Yi²  Kim, Chang Heon³ 

[1] Kongju Natl Univ, Dept Math Educ, Kong Ju 314701, South Korea

[2] Kangwon Natl Univ, Dept Math, Chunchon 200701, South Korea

[3] Sungkyunkwan Univ, Dept Math, Suwon 440746, South Korea

关键词: Sesqui-harmaonic Maass forms;    Harmonic weak Maass forms;    Traces of singular moduli;    Cycle integrals;    Regularized inner product;   

DOI  :  10.1016/j.jnt.2014.01.008

来源: Elsevier

【 摘 要 】

Borcherds -Zagier bases of the spaces of weakly holomorphic modular forms of weights 1/2 and 3/2 share the Fourier coefficients which are traces of singular moduli. Recently, Duke, Imamoglu, and Toth have constructed a basis of-the space of weight 1/2 mock modular forms, each member in which has Zagier's generating series of traces of singular moduli as its shadow. They also showed that Fourier coefficients of their mock modular forms are sums of cycle integrals of the j-function which are real quadratic analogues of singular moduli. In this paper, we prove that the Fourier coefficients of a basis of the space of weight 3/2 mock modular forms are sums of cycle integrals of a sesqui-harmonic Maass form of weight zero whose image under hyperbolic Laplacian is the j-function. Furthermore, we express these sums as regularized inner products of weakly holomorphic modular forms of weight 1/2.

【 授权许可】

Free