Conjectures on L-functions for flag bundles on Dedekind domains

Abstract

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $S:=\operatorname{Spec}(\mathcal{O}_K)$. Let $T_0,\ldots,T_n$ be regular schemes of finite type over $S$ and let $X$ be a scheme of finite type over $T_n$ with a stratification of closed subschemes (a generalized cellular decomposition) \[ \emptyset=X_{-1} \subseteq X_0 \subseteq \cdots \subseteq X_{n-1} \subseteq X_n:=X \] with $X_i-X_{i-1}=E_i$ where $E_i$ is a vector bundle of rank $d_i$ on $T_i$. We prove that if the Beilinson-Soule vanishing conjecture and Soule conjecture holds for $T_i$ it follows the same conjectures hold for $X$. We develop a criteria for the conjectures to hold in terms of an open cover and use this criteria to prove the Beilinson-Soule vanishing conjecture and Soule conjecture for the partial flag bundle $\mathbb{F}(d,E)$ of any coherent $\mathcal{O}_S$-module $E$ on $S$. Hence we get non-trivial examples where the conjectures hold in arbitrary dimension. As a special case we prove the conjectures for any affine or projective fibration of finite type over $S$. We moreover reduce the study of the Beilinson-Soule vanishing conjecture and the Soule conjecture on L-functions to the study of affine regular schemes of finite type over $\mathbb{Z}$. We also discuss the Beilinson conjecture on special values for partial flag bundles. We reduce the study of the Bloch-Kato conjecture on special values for flag bundles to the case of Dedekind domains.