Local structure of classical sequences, regular sequences, and dynamics

TL;DR

Investigates local realizability of integer sequences at primes via algebraic realizability and the Dold sign framework. Develops constructions using endomorphisms on finite $p$-groups, torsion groups $ ext{T}_p$ and locally finite products to realize $p$-parts $igl floor a_nigl floor_p$ of sequences, and uses Kummer-type congruences to control $p$-adic behaviour. Key results: the Bernoulli denominator sequence $b$ is algebraically realizable at every prime, the Bernoulli numerator sequence $t$ is realizable at regular primes and not at irregular primes, and the Euler sequence $e$ is realizable but fails at some primes (e.g., a witness at $q=61$). Shift invariance and a local-to-global viewpoint further illuminate how these sequences decompose into $p$-parts and interact with regular primes, forming a broad framework for understanding congruences and fixed-point counts in dynamical realizations.

Abstract

We introduce the notions of local realizability at a prime and algebraic realizability of an integer sequence. After discussing this notion in general we consider it for the Euler numbers, the Bernoulli denominators, and the Bernoulli numerators. This gives, for example, a dynamical characterization of the Bernoulli regular primes. Algebraic realizability of the Bernoulli denominators is shown at every prime, giving a different perspective on the great diversity of congruences satisfied by this sequence. We show that the sequence of Euler numbers cannot be realized on a nilpotent group, which may explain why it is less hospitable to congruence hunting.

Theorems & Definitions (56)

• Definition 1
• Lemma 1
• proof
• Example 2
• Example 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof