Multi-tensors of differential forms on the Hilbert modular variety and on its subvarieties
It has been shown by \textit{E. Freitag} [Math. Z. 171, 27-35 (1980; Zbl 0445.10023)] and by \textit{G. Van der Geer} [On Hilbert modular surfaces of principal congruence subgroups, Dissertation, 1977] that Hilbert modular varieties for congruence subgroups of high level satisfy some minimality property, and moreover the automorphism groups of the modular function fields are determined. The author treats the problem in the case of level one. On the other hand, there is a conjecture that subvarieties of Hilbert modular varieties are of general type if they are not too special. The aim is done if it is proven that any subvariety in the Hilbert modular variety of codimension one is of general type. The author proves it for the Hilbert modular variety \(X_ K\) of a totally real algebraic number field K in the following cases; (i) almost all k of fixed degree \(n\geq 3\), and (ii) K of degree \(n\geq 40\) with 2 \(| n\), and (iii) K of degree n with \(2\nmid n\) such that \(8\pi \tilde n/| n-4\pi -1| <1\) where \(\tilde n=\prod p^{\ell (p)+1},\) \(n- 1=(p-1) p^{\ell (p)+1} m_ p,\) \(p\nmid m_ p\), p running over the set of primes with (p-1) \(| (n-1)\). So, under the same assumptions, the Hilbert modular variety \(X_ K\) satisfies some minimality property, and the automorphism group of the modular function field over \({\mathbb{C}}\) is isomorphic to a semi-direct product \(H_ K^{(2)}\rtimes Aut(K/{\mathbb{Q}})\), where \(H_ K^{(2)}=\{x\in H_ K|\) \(x^ 2=1\}\), \(H_ K\) denoting the class group of K in the narrow sense. His method is to show the existence of enough multi-tensors of differentials extendable to a projective non-singular model of \(X_ K\) whose restrictions to any subvariety in \(X_ K\) of codimension one gives many pluri-canonical differentials on it, where \textit{K. B. Gundlach}'s estimate of weights of automorphy factors for the Hilbert modular group [Glasgow Math. J. 27, 57-80 (1985; see the preceding review)] plays an important role.
scientific article; zbMATH DE number 3920581
scientific article