Given a smooth projective family $f : X \to S$ defined over the ring of integers of a number field $K$, one obtains many different "algebraic cycle loci'' inside $S$ where the fibres of $f$ acquire additional algebraic cycles. Above the generic fibre, these loci may be studied using Hodge theory, and are commonly called "Hodge loci". The loci lying above finite primes are more elusive.

In this talk, we describe work in progress capable of studying such loci "integrally''. In particular, we describe a general criterion involving monodromy and Hodge flag level which shows that, at least away from finitely many primes of $K$, the "ordinary" positive-dimensional algebraic cycle loci at finite primes are entirely controlled by Hodge loci in characteristic zero.