Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 3

Special Issue: In Honor of Prof. Kyoji Saito’s 75th Birthday

Guest Editors: Stanislaw Janeczko, Si Li, Jie Xiao, Stephen S.T. Yau, and Huaiqing Zuo

Generalization of the Weierstrass $\wp$ function and Maass lifts of weak Jacobi forms

Pages: 371 – 420

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a3

Author

Hiroki Aoki (Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, Japan)

Abstract

Typically, a Maass lift is a map from (holomorphic) Jacobi forms of index $1$ to Siegel modular forms of degree $2$ or other kinds of modular forms. In this paper, we construct Maass lifts from weak Jacobi forms to (non-holomorphic) Siegel modular forms of degree $2$ with or without levels and characters, as formal series. By the Koecher principle, the images of our lifts are not holomorphic at cusps, even if the formal series converge. When the level is equal or less than $3$ and the character is trivial, the image of our Maass lift is in the space of meromorphic Siegel modular forms.

Keywords

$\wp$ function, Maass lifts, weak Jacobi forms

2010 Mathematics Subject Classification

Primary 11F46. Secondary 11F50.

Received 31 January 2019

Accepted 3 January 2020

Published 11 November 2020