Chow rings, cohomology rings, and point counts of moduli spaces of curves

In this expository article, we present on state-of-the art results regarding three closely related invariants of moduli spaces of curves: their Chow rings, cohomology rings, and point counts over finite fields.

We study the moduli space $\mathcal{M}_{g,n}$, parameterizing smooth genus $g$ curves with $n$ marked points, as well as its compactification by stable curves $\overline{\mathcal{M}}_{g,n}$.

After explaining the relationship between these different invariants, we survey what is know regarding the following related questions: When are the Chow rings tautological?

When are the cohomology groups tautological?

And when are the point counts over fields of size $q$ given by a polynomial in $q$?