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

2024/07/19 by Jeremy Booher, Booher, Jeremy, Jack Hsieh +9
Mathematics · #Algebraic Geometry (math.AG) #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2407.13969

openalex publication_date 2024/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Booher, Cais, Kramer-Miller and Upton study a class of 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 nth curve are given by formulae of the form α(n) p²ⁿ + β(n) pⁿ + λ_r(n) n + ν(n) , 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.

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