Point Counts of Cluster Varieties of Marked Surfaces Over Finite Fields

Abstract

We establish formulae for point counts of cluster varieties of cluster algebras of marked surfaces, possibly with punctures. We then establish formulae for the number of non-deep points in these cluster varieties over $\mathbb{F}_2$, which gives us the number of algebraic tori necessary to cover the cluster manifold over $\mathbb{F}_2$. We also show that these formulae satisfy certain recurrence relations.