Conjectures on L-functions for flag bundles on Dedekind domains

TL;DR

This paper investigates conjectures related to L-functions for flag bundles over Dedekind domains, proving their validity in new cases and reducing complex conjectures to more manageable algebraic schemes.

Contribution

It establishes conditions under which Beilinson-Soule and Soule conjectures hold for schemes with cellular decompositions, and proves these conjectures for partial flag bundles and fibrations over Dedekind domains.

Findings

Proves conjectures for partial flag bundles of coherent modules.

Reduces conjecture verification to affine regular schemes over .

Provides new examples where conjectures hold in arbitrary dimensions.

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.