Galois Groups of Ap\'ery-like Series Modulo Primes

TL;DR

This paper investigates the Galois groups of reductions modulo primes of generating series related to Apéry, Domb, and Almkvist--Zudilin numbers, revealing their behavior is influenced by prime congruences.

Contribution

It provides explicit computations of Galois groups for these series modulo primes and uncovers the role of prime congruence conditions in their behavior.

Findings

Galois groups are computed for various Apéry-like series modulo primes.

The behavior of these Galois groups depends on specific congruence conditions on p.

Prime congruences influence the structure and properties of the Galois groups.

Abstract

We compute the Galois groups of the reductions modulo the prime numbers $p$ of the generating series of Ap\'ery numbers, Domb numbers and Almkvist--Zudilin numbers. We observe in particular that their behavior is governed by congruence conditions on p.