Higher a-numbers in ${\mathbf{Z}}_p$-towers via Counting Lattice Points

Booher, Cais, Kramer-Miller and Upton study a class of ${\mathbf{Z}}_p$-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 $n$th curve are given by formulae of the form $\alpha (n)p^{2n}+\beta (n)p^n+{\lambda }_r(n)n+\nu (n)$, where $\alpha ,\beta ,\nu ,{\lambda }_r$ 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.