The Artin-Hasse p-section: weighted convolutions and p-adic recovery
Abstract
Let p be an odd prime, and let an be the reduction modulo p of the nth coefficient of the Artin-Hasse exponential. We study the weighted p-section convolutions Wk formed from the coefficients akp. We prove the conjecture of Avitabile and Mattarei for 1<k<p and extend it to a single global power-series identity determining the entire sequence (Wk)k≥ 0. This identity yields an explicit base-p digit formula; in particular, (Wk) is p-automatic and admits an explicit finite-state evaluator. Independently, the exact conjugated p-section equation gives a strict p-adic fixed-point iteration for the logarithmic derivative of the p-section. Big-Witt reconstruction then recovers the p-section, and hence all coefficients of the Artin-Hasse exponential, to arbitrary prescribed p-adic precision by a purely modular procedure.
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BibTeX
@misc{clare2026artinhasse,
title = {{The Artin-Hasse $p$-section: weighted convolutions and $p$-adic recovery}},
author = {Ben Clare},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07761},
eprint = {2609.07761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07761},
}
RIS
TY - UNPB
TI - The Artin-Hasse p-section: weighted convolutions and p-adic recovery
AU - Ben Clare
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07761
UR - https://arxiv.org/abs/2609.07761
AB - Let p be an odd prime, and let a_n be the reduction modulo p of the nth coefficient of the Artin-Hasse exponential. We study the weighted p-section convolutions W_k formed from the coefficients a_kp. We prove the conjecture of Avitabile and Mattarei for 1<k<p and extend it to a single global power-series identity determining the entire sequence (W_k)_k≥ 0. This identity yields an explicit base-p digit formula; in particular, (W_k) is p-automatic and admits an explicit finite-state evaluator. Independently, the exact conjugated p-section equation gives a strict p-adic fixed-point iteration for the logarithmic derivative of the p-section. Big-Witt reconstruction then recovers the p-section, and hence all coefficients of the Artin-Hasse exponential, to arbitrary prescribed p-adic precision by a purely modular procedure.
ER -