On the local zeta functions of the Hilbert modular schemes
Similar to the Siegel case the author formulates results without details or proofs of an action \(f_{\bullet}\) of the Hecke ring HR(\(\Gamma\) (N), \(G_ 2^+({\mathcal O}_ K))\) of a totally real algebraic number field K of degree g with ring of integers \({\mathcal O}_ K\) on the singular homology \(H_{\bullet}(M(N)_{{\mathbb{C}}},{\mathbb{Z}})\) of a smooth toroidal compactification \(M(N)_{{\mathbb{C}}}\) of the Hilbert modular variety \(\Gamma (N)\setminus {\mathbb{H}}^ g\), \(\Gamma\) (N) being the principal congruence subgroup of level \(N\geq 3\) of the Hilbert modular group of K as well as an \(\ell\)-adic version \(f^{\bullet}\) for the \(\ell\)-adic cohomology of \(M(N)_{{\mathbb{C}}}\) resp. D(N). Here D(N) is a suitable proper smooth Hilbert modular variety over the field \(\bar F_ p\), \(p\nmid N\) prime. The main results are (1) an estimation of the eigenvalues \(\lambda_ p\) of the Hecke operator \(f^ n(Tp\) \({\mathcal O}_ K):\) \[ | \lambda_ p| \leq p^{n/2}+p^{(2g-n)/2}\text{ for } 0\leq n\leq 2g, \] (2) an expression of the local zeta function Z(D(N),X) in terms of the above Hecke operators: \[ Z(D(N),X)=\sqrt{\prod_{n}P^*_ n(M(N)_{{\mathbb{C}}},X)^{(-1)^{n+1}}} \] where the polynomial \(P^*_ n\) equals \[ \det [1-(f_ n(Tp\quad {\mathcal O}_ K)\otimes id)X+(f_ n(\Gamma (N)\sigma_ p\Gamma (N))\otimes id)p^ gX^ 2]. \]
scientific article; zbMATH DE number 4191781