Higher a-numbers in Zₚ-towers via Counting Lattice Points

Booher, Cais, Kramer-Miller and Upton study a class of Zₚ-tower of curves in characteristic p with ramification controlled by an integer d. In the special case that d divides $p-1$, they prove a formula for the higher a-numbers of these curves involving the number of lattice points in a complicated region of the plane. Booher and Cais had previously conjectured that for n sufficiently large the higher a-numbers of the nth curve are given by formulae of the form α(n) p²ⁿ + β(n) pⁿ + λᵣ(n) n + ν(n) for n sufficiently large, where α,β,ν,λᵣ are periodic functions of n. This is an example of a new kind of Iwasawa theory. We establish this conjecture by carefully studying these lattice points.