Galois Groups of Apéry-like Series Modulo Primes

TL;DR

This work determines the Galois groups of reductions modulo primes for the generating series of Apéry-type sequences, notably Apéry, Domb, and Almkvist–Zudilin numbers, via their $p$-Lucas algebraic relations. By exploiting a modular parameterization and a square-substitution framework $f(t)=(1+x)h(x)^2$, the authors reduce the problem to analyzing abelian extensions and prolongations of automorphisms, yielding a clean residue-based dichotomy: the Galois group is the subgroup $S$ of squares when certain primes lie in specific congruence classes modulo 24, and the full multiplicative group otherwise; this pattern is encoded in the factorization of the truncation $A_p(t)$. The approach extends to Apéry-like sequences and to a broader “zoo” of Zagier’s sporadic examples, with analogous square-factorization phenomena and modular forms connections via Hauptmoduln. Collectively, the results provide evidence for uniformity phenomena in the Galois groups of reductions of D-finite series and illuminate deep arithmetic structures tied to modular parameterizations and hypergeometric functions.

Abstract

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

Figures (2)

• Figure 1: Zagier's sporaric examples Zag09AVZ11
• Figure 2: Examples of sequences connected to modular forms

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof