Theory of newforms of half-integral weight

Murugesan Manickam, Jaban Meher and Balakrishnan Ramakrishnan

Vol. 274 (2015), No. 1, 125–139

DOI: 10.2140/pjm.2015.274.125

Abstract

We set up the theory of newforms of half-integral weight on ${\Gamma }_0\left(8N\right)$ and ${\Gamma }_0\left(16N\right)$, where $N$ is odd and squarefree. Further, we extend the definition of the Kohnen plus space in general for trivial character and also study the theory of newforms in the plus spaces on ${\Gamma }_0\left(8N\right)$, ${\Gamma }_0\left(16N\right)$, where $N$ is odd and squarefree. Finally, we show that the Atkin–Lehner $W$-operator $W_4$ acts as the identity operator on $S_{2k}^{new}\left(4N\right)$, where $N$ is odd and squarefree. This proves that $S_{2k}^{-}\left(4\right)=S_{2k}\left(4\right)$.

Keywords

modular forms of half-integral weight, theory of newforms, Kohnen plus space

Mathematical Subject Classification 2010

Primary: 11F37

Secondary: 11F11, 11F32

Milestones

Received: 24 April 2014

Revised: 22 August 2014

Accepted: 28 August 2014

Published: 2 March 2015