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

TL;DR

The article studies higher $a$-numbers $a^{r}(X_n)$ in $oldsymbol{Z}_p$-towers of curves over a field of characteristic $p$, with ramification invariant $d$. For the case $d ig| p-1$, it proves an exact lattice-point formula that expresses $a^{r}(X_n)$ as a leading quadratic term in $p^n$ plus a linear term in $n$ and a periodic correction $ u_r(n)$, aligning with and strengthening Booher–Cais-type conjectures. The method reinterprets $a^{r}(X_n)$ as the count of lattice points in a scaled triangle, and then analyzes floor-sum and boundary contributions via base-$p$ expansions and a delta-boundary indicator, yielding precise periodicity properties. These results demonstrate a new Iwasawa-theoretic phenomenon for higher $a$-numbers in odd characteristic, where the growth in genus drives regular, structured fluctuations rather than the classical $p$-rank phenomena. The paper also clarifies the period structure, showing the minimal period is controlled by the order of $p$ modulo certain denominators, and provides concrete examples and corollaries for small primes.

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 $α(n) p^{2n} + β(n) p^n + λ_r(n) n + ν(n) $ for $n$ sufficiently large, where $α,β,ν,λ_r$ 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.

Figures (2)

• Figure 1: $\widetilde{\Delta}_2$ and $\mathcal{P}_2$ (shaded) when $p=5$, $r=2$, and $d=4$.
• Figure 2: $\widetilde{\Delta}_2$ and $\mathcal{P}_2$, showing $\mu_i$, when $p=7$, $r=2$, and $d=3$.

Theorems & Definitions (82)

• Definition 1.1
• Conjecture 1.2: Booher-Cais boohercais
• Remark 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6: Booher, Cais, Kramer--Miller, and Upton
• proof
• Theorem 1.7
• Example 1.8
• Example 1.9