The action of modular operators on the Fourier--Jacobi coefficients of modular forms

V. A. Gritsenko

Sbornik. Mathematics, Tome 47 (1984) no. 1, pp. 237-268

Résumé

The author studies the imbedding of the Hecke $p$-ring $L_p^{n+1}$ of the modular group $\mathrm{Sp}_{n+1}(\mathbf{Z})$ of genus $n+1$ in the Hecke ring $L_p^{n,1}$ of the group $\Gamma_{n,1}$ given by $$ \Gamma_{n,1}=\left\{\begin{pmatrix} A0*\\ ****\\ C0*\\ 000* \end{pmatrix}\in\mathrm{Sp}_{n+1}(\mathbf{Z})\right\}. $$ It is proved that the Hecke polynomial $Q_{n,1}^{(n+1)}(z)$ of $L_p^{n+1}$ splits over $L_p^{n,1}$, and the coefficients of the factors can be written explicitly in terms of the coefficients of the Hecke polynomial $Q^{(n)}(z)$ of genus $n$ and “negative” powers of a particular element $\Lambda$ of $L_p^{n,1}$. The "$-1$ power" of $\Lambda$ is computed and a formula for $\Lambda^{-2}$ is presented. The results that are obtained permit one to describe a large class of power series constructed from the Fourier–Jacobi coefficients by means of eigenfunctions with denominators depending only on the eigenvalues. Bibliography: 19 titles.