Saito–Kurokawa liftings of level N and practical construction of Jacobi forms

Abstract

Saito–Kurokawa liftings from Jacobi forms of degree one of weight $k$ of level $N$ to Siegel modular forms of degree two of weight $k$ of level $N$ with or without character are explicitly given by describing their Fourier expansions. Their $L$-functions are also given as well as the action of Hecke operators including some operators at bad primes. Also, a practical way of constructing Jacobi forms of given level is explained. This is explicitly executed for Jacobi forms of level up to $5$, and their explicit structure theorems are given.