We contribute to the arithmetic/topology dictionary by relating asymptotic point counts and arithmetic statistics over finite fields to homological stability and representation stability over C in the example of configuration spaces of n points in smooth varieties. To do this, we import the method of homological stability from the realm of topology into the theory ofétale cohomology; in particular we give the first examples of stability ofétale cohomology groups as Galois representations where the Galois actions are not already explicitly known. We then establish subexponential bounds on the growth of the unstable cohomology, and we apply this andétale homological stability to compute the large n limits of various arithmetic statistics of configuration spaces of varieties over F q .
In REF Theorem B] they prove that configuration spaces of n points in smooth varieties have convergent cohomology REF Definition 3 .1] and obtain arithmetic statistics for configuration spaces over finite fields REF Theorem C] .
\'Etale homological stability and arithmetic statistics
arXiv: Algebraic Geometry
Mathematics