论文标题
在Dedekind域上的旗帜捆绑包上的猜想
Conjectures on L-functions for flag bundles on Dedekind domains
论文作者
论文摘要
令$ \ MATHCAL {O} _K $为代数数字字段$ K $中的整数环，让$ s：= \ operatatorName {spec}（\ Mathcal {o} _K）$。令$ t_0，\ ldots，t_n $为$ s $以上的有限类型的常规方案，让$ x $是$ t_n $的有限类型的方案，并分层封闭的子机（广义蜂窝分解）\ [\ emptyset = x___________________________________________________________________ \ subseteq x_n：= x \]，带有$ x_i-x_ {i-1} = e_i $，其中$ e_i $是$ t_i $上的级别$ d_i $的向量捆绑包。我们证明，如果Beilinson-Soule消失的猜想和Soule的猜想对$ T_I $保持，则遵循$ x $的同一猜想。我们制定了一个猜想的标准，可以在开放式封面方面坚持下去，并使用此标准证明了贝林森 - 苏（Beilinson-soule）消失的猜想和soule猜想的偏旗捆绑$ \ mathbb {f}（f}（d，e）$的任何相干$ \ mathcal $ \ nathcal {o} o} o} _s $ _s $ $ e $ $ e $ $ $ $ $ e $ o。因此，我们得到了猜想在任意维度中的非平凡示例。作为特殊情况，我们证明了有限类型的任何仿度或投影纤维的猜想。此外，我们还减少了对贝林森 - 苏（Beilinson-Soule）消失的猜想的研究，以及对L功能的Soule猜想，研究有限类型的仿射常规方案超过$ \ Mathbb {Z} $。我们还讨论了贝林森关于部分旗帜束的特殊值的猜想。我们减少了Bloch-Kato对Dedekind域的特殊值的猜想的研究。
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.
