Gritsenko lift (awaiting review)

Given a Jacobi cusp form $\phi(\tau,z)\in J_{k,N}^{\text{cusp}}$, the Gritsenko lift of $\phi$,

$$Grit(\phi)\in S_k(K(N)),$$

is a paramodular form defined by the formula on its Fourier coefficients

$$a(Grit(\phi); n,r,m) = \sum_{d|(n,r,m)} d^{k-1} c(\phi; mn/d^2, r/d).$$

If $\phi$ is a Jacobi cusp form, then $Grit(\phi)$ is a paramodular cusp form.

The L-function of the Gritsenko lift is related to the L-function of an elliptic modular form that is related to the Jacobi form.