Counting mod n in pseudofinite fields

Abstract

We show that in an ultraproduct of finite fields, the mod-$n$ nonstandard size of definable sets varies definably in families. Moreover, if $K$ is any pseudofinite field, then one can assign "nonstandard sizes mod $n$" to definable sets in $K$. As $n$ varies, these nonstandard sizes assemble into a definable strong Euler characteristic on $K$, taking values in the profinite completion $\hat{\mathbb{Z}}$ of the integers. The strong Euler characteristic is not canonical, but depends on the choice of a nonstandard Frobenius. When $\operatorname{Abs}(K)$ is finite, the Euler characteristic has some funny properties for two choices of the nonstandard Frobenius. Additionally, we show that the theory of finite fields remains decidable when first-order logic is expanded with parity quantifiers. However, the proof depends on a computational algebraic geometry statement whose proof is deferred to a later paper.