论文标题
积分级别的代数周期基因座
Algebraic Cycle Loci at the Integral Level
论文作者
论文摘要
令$ f：x \ to s $为一个平稳的投影家族，定义在$ \ Mathcal {o} _ {k} [\ Mathcal {s}^{ - 1}] $中，其中$ k \ subset \ subset \ subset \ mathbb {c} $ is a number field和$ \ \ \ \ \ \ \ \ \ \ mathcal {s} $是prime的prime prime。对于每个Prime $ \ Mathfrak {p} \ in \ Mathcal {o} _ {k} [\ Mathcal {s}^{ - 1}] $，带有残基$κ（\ Mathfrak {p}）$哪个共同学循环猜想预测了代数周期的非平凡家族的存在，从而推广了通用纤维$ s _ {\ overline {k}} $的hodge基因座。我们开发了一种在整体级别一起研究所有此类基因座的技术。结果，我们为非平凡的普通代数周期基因座的结合提供了非Zariski密度标准。该标准是相当笼统的，仅取决于固定的共同体学位$ W $和相关几何单肌表示的Zariski密度的Hodge Flag的水平。
Let $f : X \to S$ be a smooth projective family defined over $\mathcal{O}_{K}[\mathcal{S}^{-1}]$, where $K \subset \mathbb{C}$ is a number field and $\mathcal{S}$ is a finite set of primes. For each prime $\mathfrak{p} \in \mathcal{O}_{K}[\mathcal{S}^{-1}]$ with residue field $κ(\mathfrak{p})$, we consider the algebraic loci in $S_{\overline{κ(\mathfrak{p})}}$ above which cohomological cycle conjectures predict the existence of non-trivial families of algebraic cycles, generalizing the Hodge loci of the generic fibre $S_{\overline{K}}$. We develop a technique for studying all such loci, together, at the integral level. As a consequence we give a non-Zariski density criterion for the union of non-trivial ordinary algebraic cycle loci in $S$. The criterion is quite general, depending only on the level of the Hodge flag in a fixed cohomological degree $w$ and the Zariski density of the associated geometric monodromy representation.
