Siegel modular forms of half integral weight and a lifting conjecture

Abstract

A conjecture on lifting to Siegel cusp forms of half-integral weight $k - 1/2$ of degree two from each pair of cusp forms of $SL_{2}(\mathbb{Z})$ of weight $2k - 2$ and $2k - 4$ is given with a conjectural relation of the $L$ functions and numerical evidences. We also describe the space of Siegel modular forms of half-integral weight, its “plus subspace” and Jacobi forms of degree two by explicitly given theta functions.