On the action of Hecke rings on homology groups of smooth compactifications of Siegel modular varieties and Siegel cusp forms (Q752744)

Let \(\Gamma\) be the principal congruence subgroup of level \(N\geq 3\) of Siegel's modular group Sp(g,\({\mathbb{Z}}),M\) a projective smooth toroidal compactification of the quotient \(\Gamma \setminus {\mathbb{H}}_ g\) where \({\mathbb{H}}_ g\) is nothing but Siegel's upper half plane of degree \(g\geq 2\) and \(H^{p,q}(M)\) the space of harmonic forms of type (p,q) with respect to a real analytic, Sp(g,\({\mathbb{Z}})\)-invariant Hodge metric which by projectivity of M always exists. Then the finite abelian group \(\Gamma '/\Gamma\), \(\Gamma '=\{\xi \in Sp(g,{\mathbb{Z}}) |\xi\) mod N diagonal\(\}\), operates on the harmonic forms, thus inducing a direct decomposition \(H^{p,q}(M)=\oplus H^{p,q}(\chi,M)\) in invariant subspaces \[ H^{p,q}(\chi,M)=\{\phi \in H^{p,q}(M)| \gamma^*\phi =\chi (\gamma mod \Gamma)\phi \text{ for all } \gamma \in \Gamma '\} \] where the summation is taken over all characters \(\chi \in (\Gamma '/\Gamma)^*.\) On the other hand there is a natural ring-homomorphism \(f_ m\) of the Hecke ring \(HR(\Gamma,GSp^+(g,{\mathbb{Z}}))\) into the endomorphism ring of the singular homology group \(H_ m(M)\), introduced by the author in a previous paper. Then for all \((n,N)=1\) the Hecke operator T(n) gives rise to an endomorphism \(^ t(f_{p+q})(T(n))\otimes_{{\mathbb{Z}}}id)\) in cohomology, leaving all the spaces \(H^{p,q}(\chi,M)\) invariant (Th. 1). Restricting to the case \(g=2\) one gets in addition an estimation of an eigenvalue \(\lambda_ n\) of \(f_ 3(T(n)\otimes id)\) which in case n prime, \(n\nmid N\) takes the form \(| \lambda_ n| <(1+n)(1+n^ 2)\) (Th. 2). As to the eigenvalues \(\lambda\) (n) of the usual Hecke operators \(T_{g+1+w}(n)\) acting on the cusp forms \(S_{g+1+w}(\Gamma)\) of weight \(g+1+w\) the author gets a similar estimation, thus improving and surpassing a well known result. Again for n prime, n \(\nmid N\) one gets \[ | \lambda (n)| <n^{gw/2}(\prod^{g}_{k=1}(1+n^ k))\quad (Th.\quad 3). \] (Equality does not hold.) All the proofs stem heavily on former studies of the author where especially an interpretation of \(^ t(f_{p+q}(T(n)\otimes_{{\mathbb{Z}}}id)\) as integral operator H coming from the orthogonal projection in potential theory \(id=H+d\delta G+\delta dG\) is given. Additionally in the case \(g=2\) the argumentation is based on the vanishing of \(H^ 1(M,{\mathbb{C}})\) [see also the reviewer, Abh. Math. Sem. Univ. Hamburg 57, 203-213 (1987; Zbl 0617.32045)] and the Hodge-Signature-Theorem.