Siegel Eisenstein series of arbitrary level and theta series
Let \(f\) be a Siegel modular form of genus \(n\) and level \(q\) which does not vanish at all zero-dimensional cusps. Generalizing an observation of the reviewer, the author shows that \(f\) is uniquely determined by its values at the zero-dimensional cusps if the weight of \(f\) is large enough and if \(f\) is an eigenfunction of the Hecke operator \((T(p))^m\) for some \(m\geq 1\) and for at least one prime number \(p\equiv\pm 1\bmod q\). This implies that each \(f\) of the type above is a linear combination of the transformed Eisenstein series \(E_r\mid L\) \((L\in\text{Sp} (n,\mathbb{Z})\); \(r\) sufficiently large). In particular, the Siegel mean value \(f\) of the theta series of the genus of a positive definite even integral quadratic form satisfies the assumptions of the above theorem, and this yields a new proof of (a restricted version) of the analytic form of Siegel's Hauptsatz. The author also investigates the converse problem to express Eisenstein series as linear combinations of theta series and proves; A linear combination \(E\) of Eisenstein series can be expressed as a linear combination of suitable theta series if and only if the values of \(E\) at the zero-dimensional cusps satisfy the same relations which are a priori known to hold for the theta functions.
scientific article; zbMATH DE number 977868