Algebraic Cycle Loci at the Integral Level

Abstract

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 $\kappa(\mathfrak{p})$, we consider the algebraic loci in $S_{\overline{\kappa(\mathfrak{p})}}$ 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.