We compute the p-adic densities of points with a given splitting type along a finite map, analogous to the classical Chebotarev theorem over number fields and function fields. Under certain niceness hypothesis, we prove that these densities satisfy a functional equation in the size of the residue field. As a consequence and building on previous work of one of us, we prove a conjecture of Gajovic, Cremona, Bhargava and Fisher in the tame case. The key tool is the notion of an admissible pair associated to a group, which we use as an invariant of the inertia and decomposition action of a local field on the fibers of the finite map. We compute the splitting densities by an inductive argument involving these invariants and p-adic integration.

We present techniques for computing motivic measures of configuration spaces when points of the base space are weighted. This generality allows for a more flexible notion of configuration space which can be useful in applications, and subsumes some disparate problems that were previously given ad-hoc treatments. The space of effective zero-cycles of given weight generalizes the ordinary symmetric product and has a canonical stratification by splitting type. We prove formulas for the motivic measures of open and closed strata of the space of zero-cycles of given weight on a quasiprojective variety with an arbitrary grading by natural numbers. This lets us compute, among other things, the motive of the space of geometrically irreducible hypersurfaces in projective space. We introduce a general form of plethysm suitable for the graded situation which makes use of a new generalization of symmetric functions. Our main formula solves a motivic formulation of the inverse problem to Pólya enumeration.

For varieties over a finite field with “many” automorphisms, we study the ℓ-adic properties of the eigenvalues of the Frobenius operator on their cohomology. The main goal of this paper is to consider “geometric Iwasawa towers” and prove that the characteristic polynomials of the Frobenius on the étale cohomology show a surprising ℓ-adic convergence. We prove this by proving a more general statement about the convergence of certain invariants related to a skew-abelian cohomology group. Along the way, we will prove that many natural sequences converge ℓ-adically and give explicit rates of convergence. In a different direction, we provide a precise criterion for curves with many automorphisms to be supersingular, generalizing and unifying many old results.

Given two varieties V,W in the n-fold product of modular curves, we answer affirmatively a question (formulated by Shou-Wu Zhang’s AIM group) on whether the set of points in V that are Hecke translations of some point on W is dense in V. We need to make some (necessary) assumptions on the dimensions of V,W but for instance, when V is a divisor and W is a curve, no further assumptions are needed. We also examine the necessity of our assumptions in the case of unlikely intersections and show that, contrary to exceptions, two curves in a high dimensional space over a finite field can intersect infinitely often up to Hecke translations.

We prove an intersection-theoretic result pertaining to curves in certain Hilbert modular surfaces in positive characteristic. Specifically, we show that given two appropriate curves C,D parameterizing abelian surfaces with real multiplication, the set of points in (x,y)∈C×D with surfaces parameterized by x and y isogenous to each other is Zariski dense in C×D, thereby proving a case of a just-likely intersection conjecture. We also compute the change in Faltings height under appropriate p-power isogenies of abelian surfaces with real multiplication over characteristic p global fields.

In this short note, we prove a criteria for supersingularity when the variety has a large automorphism group and a perfect bilinear pairing. This criteria unifies and extends many known results on the supersingularity of curves and varieties and in particular, applies to a large family of Artin-Schreier curves.