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

TL;DR

This paper proves a conjecture relating higher a-numbers in certain $ extbf{Z}_p$-tower curves to lattice point counts, revealing a new Iwasawa-theoretic pattern in characteristic p.

Contribution

It establishes a conjecture connecting higher a-numbers to lattice point counts and introduces a novel Iwasawa theory framework for these curves.

Findings

Higher a-numbers are expressed via lattice point counts.

Confirmed the conjectured formula for large n.

Identified periodic functions governing the a-number growth.

Abstract

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.