Bases of spaces of harmonic weak Maass forms and Shintani lifts of harmonic weak Maass forms
- PDF / 400,018 Bytes
- 26 Pages / 439.37 x 666.142 pts Page_size
- 77 Downloads / 201 Views
Bases of spaces of harmonic weak Maass forms and Shintani lifts of harmonic weak Maass forms Daeyeol Jeon1 · Soon-Yi Kang2 · Chang Heon Kim3 Received: 25 February 2019 / Accepted: 26 July 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We construct bases of the space of harmonic weak Maass forms of weight κ ∈ 21 Z. Using these bases, we obtain a Shintani lift from a positive integral weight harmonic weak Maass form to a half-integral weight harmonic weak Maass form, which reduces to the classical Shintani lift on the space of cusp forms. Keywords Weakly holomorphic modular forms · Harmonic weak Maass forms · Regularized inner products · Shintani lifts Mathematics Subject Classification 11F03 · 11F12 · 11F30 · 11F37
1 Introduction In the beginning of the new millennium, two important but seemingly unrelated results were discovered. Zwegers [39] solved an old puzzle on Ramanujan’s mock theta functions by showing that they can be completed to real analytic modular forms, and Zagier [38] proved that the traces of singular moduli are Fourier coefficients of a
Daeyeol Jeon was supported by the National Research Foundation of Korea (NRF) grant (NRF-2016R1D1A1B03934504). Soon-Yi Kang was supported by the NRF grant (NRF-2016R1D1A1B01012258). Chang Heon Kim was supported by the NRF grant (NRF-2018R1D1A1B07045618 and 2016R1A5A1008055).
B
Chang Heon Kim [email protected] Daeyeol Jeon [email protected] Soon-Yi Kang [email protected]
1
Department of Mathematics Education, Kongju National University, Gongju 32588, Korea
2
Department of Mathematics, Kangwon National University, Chuncheon 24341, Korea
3
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
123
D. Jeon et al.
weight 3/2 weakly holomorphic modular form. Almost concurrently, Bruinier and Funke [11] defined harmonic weak Maass forms as theta lifts. This work attracted enormous attention as it was found that the aforementioned discoveries belong to the theory of harmonic weak Maass forms. In fact, Zweger’s real analytic modular forms are examples of harmonic weak Maass forms [4] and the traces of CM values of a weakly holomorphic modular function of arbitrary level are Fourier coefficients of the holomorphic part of a weight 3/2 harmonic weak Maass form [12]. (For further details on the theory, refer to [10].) In particular, the arithmetic properties of Fourier coefficients of harmonic weak Maass forms related to traces of singular moduli give a connection between the spaces of harmonic weak Maass forms of integral and half-integral weight. In order to elucidate this phenomenon, we need to introduce some notation. Throughout κ ∈ 21 Z, ν ∈ N, and D is a fundamental discriminant. For convenience, we let (1) = SL2 (Z), if κ ∈ Z, := 0 (4), if κ ∈ 21 + Z and let Hκ! ⊇ Mκ! ⊇ Sκ denote the spaces of harmonic weak Maass forms, weakly holomorphic modular forms and cusp forms on of weight κ, respectively. When ! κ ∈ 21 + Z, we assume the elements in H κ satisfy Kohnen’s plus space condition; that is,
Data Loading...
