On~the connection of the eigenvalues of~Hecke operators and the Fourier coefficients of eigenfunctions for Siegel's modular forms of genus~$n$

N. A. Zharkovskaya

Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 549-557

Résumé

Let $f(z)=\sum_{N\geqslant0}a(N)\exp2\pi i\sigma(NZ)$ be Siegel's modular form of genus $n$ which is an eigenfunction for all operators in the $p$-component of a Hecke ring; in particular, $T_{p^\delta}f(Z)=\lambda_f(p^\delta)f(Z)$. This paper examines the series $\sum_{\delta=0}^\infty a(p^\delta N)t^\delta$ ($p$ does not divide $N$). It is proved that each such series is a rational function, where the degree of the numerator of this function does not exceed $2^n-2$ and the denominator coincides with the denominator of the series $\sum_{\delta=0}^\infty \lambda_f(p^\delta)t^\delta$. Bibliography: 6 titles.