Truncated versions of Dwork's lemma for exponentials of power series and $p$-divisibility of arithmetic functiens

TL;DR

This paper extends Dwork's lemma to establish new p-adic valuation bounds for exponential power series coefficients, leading to results on p-divisibility and periodicity of permutation and subgroup numbers.

Contribution

It introduces weaker conditions for p-adic valuation bounds of exponential power series, generalizing previous results and applying them to permutation representations and subgroup enumeration.

Findings

Derived lower bounds on p-adic valuations of exponential coefficients.

Established conditions for periodicity of subgroup numbers modulo p.

Proved a supercongruence for a specific binomial sum.

Abstract

(Dieudonn\'e and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the coefficients of the exponential of $S(z)$, assuming weaker conditions on the coefficients of $S(z)$ than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the $p$-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbitrary finite Abelian $p$-group, thus generalising numerous results in special cases that had appeared earlier in the literature. Further applications include sufficient conditions for ultimate periodicity of subgroup numbers modulo $p$ for free products of finite Abelian $p$-groups, results on $p$-divisibility of permutation numbers with restrictions on their cycle structure, and a curious "supercongruence" for a certain binomial sum.