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Galois Groups of Apéry-like Series Modulo Primes

This paper computes the Galois groups of the reductions modulo primes of the generating series for Apéry, Domb, and Almkvist–Zudilin numbers, demonstrating that their behavior is determined by specific congruence conditions on the prime $p$ .

Original authors: Xavier Caruso, Florian Fürnsinn, Daniel Vargas-Montoya, Wadim Zudilin

Published 2026-02-18

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CC BY 4.0

Original authors: Xavier Caruso, Florian Fürnsinn, Daniel Vargas-Montoya, Wadim Zudilin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, infinite recipe book. Each page of this book contains a specific number, and if you follow the rules to generate the next number, you get a sequence that grows in a very predictable, yet mysterious way. Mathematicians call these Apéry-like numbers. They are famous because one of them helped prove that a specific mathematical constant (related to the volume of a sphere in higher dimensions) cannot be written as a simple fraction.

Now, imagine you want to study these numbers, but instead of looking at the whole infinite book, you decide to play a game of "modulo arithmetic." Think of this as looking at the numbers through a special pair of glasses that only shows you the remainder when you divide by a specific prime number (like 3, 5, 7, 11, etc.).

This paper is about what happens when you look at these infinite sequences through the "modulo glasses" of different prime numbers. The authors discovered that the behavior of these sequences isn't random; it follows a strict, hidden code based on the prime number you chose.

Here is the breakdown of their discovery using simple analogies:

1. The "Shadow" of the Sequence

When you look at these infinite sequences modulo a prime number $p$ , something magical happens: the infinite series suddenly stops being infinite and becomes a finite polynomial (a short algebraic expression).

Think of the infinite sequence as a long, winding river. When you look at it through the "modulo $p$ " glasses, the river seems to flow into a small, contained pond. The shape of this pond is determined by a specific polynomial, let's call it $A_p$ .

2. The Square Root Mystery

The central question the authors asked was: "Is this pond a perfect square?"

In math, if you can write a polynomial as something squared (like $(x+1)^2$ ), it's called a "square." If you can't, it's not.

The Discovery: The authors found that whether the pond is a perfect square or not depends entirely on the prime number $p$ and a specific "congruence condition" (basically, what remainder $p$ leaves when divided by a specific number, like 24 or 8).

The Analogy: Imagine you have a magic lock.

If the key (the prime number) is a certain type (e.g., leaves a remainder of 1, 5, 7, or 11 when divided by 24), the lock opens perfectly, and the shape is a perfect square.
If the key is a different type (e.g., leaves a remainder of 13, 17, 19, or 23), the lock is slightly broken. The shape is almost a square, but it has a "twist" or a "knot" in it (represented by a specific extra factor like $t^2 - 34t + 1$ ).

3. The "Galois Group" (The Security Guard)

The paper talks a lot about Galois Groups. In simple terms, think of the Galois Group as a security guard or a bouncer at a club.

The "club" is the mathematical world created by the sequence.
The "bouncer" decides who gets in and how they can move around.
The authors calculated exactly how strict this bouncer is for different prime numbers.

They found that the bouncer's behavior is uniform.

Scenario A: For some primes, the bouncer is very strict. He only allows people who are "squares" to enter. The group of allowed people is smaller.
Scenario B: For other primes, the bouncer is more relaxed. He lets everyone in, regardless of whether they are squares. The group is larger.

The paper proves that you can predict exactly which bouncer (strict or relaxed) you will get just by looking at the prime number's "remainder signature."

4. The Three Main Characters

The paper focuses on three specific "families" of these number sequences:

Apéry Numbers: The famous ones.
Domb Numbers: The "alternating" cousins.
Almkvist–Zudilin Numbers: The "signed" cousins.

Even though they look different on the surface, the authors discovered they all dance to the same rhythm. When you apply the "modulo glasses," they all reveal the same pattern: their shapes (the polynomials) are either perfect squares or squares with a specific twist, depending on the prime number.

5. The "Hidden Map" (Modular Parameterization)

How did they solve this? They used a "secret map."
They realized that these complex number sequences are actually just shadows of a much simpler, underlying function (related to a function called $h$ ).

Imagine the complex sequence is a complicated 3D sculpture.
The authors found a way to project that sculpture onto a 2D wall (using a variable $x$ ).
On this 2D wall, the sculpture looks like a simple square ( $h^2$ ).
The "twist" or the "extra factor" they found in the polynomials comes from how the 3D sculpture is rotated or stretched when projected onto the wall.

Why Does This Matter?

You might ask, "Who cares if a polynomial is a square modulo 7?"

This is important because it connects two very different worlds of math:

Number Theory: The study of integers and primes.
Differential Equations: The study of how things change and flow (calculus).

The paper suggests a deep, hidden rule: The way these numbers behave modulo primes is a "shadow" of the way they behave in continuous calculus.

It's like noticing that the pattern of raindrops hitting a puddle (discrete math) perfectly matches the ripples created by a stone thrown in a lake (continuous math). The authors proved that this isn't a coincidence; it's a fundamental law of these specific number sequences.

Summary

In short, this paper is a detective story. The authors investigated three famous families of numbers, looked at them through the lens of different prime numbers, and discovered that their "shapes" (Galois groups) follow a strict, predictable code. Whether the shape is a perfect square or a twisted square depends entirely on the prime number's remainder when divided by 24 (or 8, or 6). They proved this using a clever mathematical "projection" that simplifies the complex into the understandable.

1. Problem Statement

The paper investigates the arithmetic properties of generating series associated with specific integer sequences known as Apéry-like numbers (specifically Apéry numbers, Domb numbers, and Almkvist–Zudilin numbers) when reduced modulo a prime number $p$ .

Context: These sequences are famous for their role in irrationality proofs (e.g., Apéry's proof of the irrationality of $\zeta(3)$ ) and their connection to modular forms. Their generating functions, $f(t)$ , are D-finite (satisfy linear differential equations with polynomial coefficients).
The Phenomenon: For these sequences, the coefficients satisfy the $p$ -Lucas property: $\alpha_{np+\ell} \equiv \alpha_n \alpha_\ell \pmod p$ . This implies a functional congruence for the generating series:
$f(t) \equiv A_p(t) \cdot f(t)^p \pmod p$
where $A_p(t)$ is the truncation of $f(t)$ modulo $p$ up to degree $p-1$ .
The Core Question: Since $A_p(t)$ is algebraic over $\mathbb{F}_p(t)$ , the extension $\mathbb{F}_p(t, f) / \mathbb{F}_p(t)$ is a Kummer extension. The authors aim to determine the Galois group of this extension for various primes $p$ . Specifically, they seek to classify whether the Galois group is the full multiplicative group $\mathbb{F}_p^\times$ or its unique subgroup of index 2 (the squares, denoted $S$ ), and how this classification depends on congruence conditions on $p$ .

2. Methodology

The authors employ a combination of algebraic geometry, Kummer theory, and hypergeometric function theory.

Rational Parameterization: The central technique involves relating the generating series $f(t)$ $f (t)$ to the generating function of the Franel numbers, $h(x) = \sum (\binom{n}{k})^3 x^n$ $h (x) = \sum ((k n))^{3} x^{n}$ .
- For Apéry numbers: $f_\alpha = (1+x)h(x)^2$ with $t = \frac{x(1-8x)}{1+x}$ .
- Similar rational substitutions exist for Domb and Almkvist–Zudilin numbers.
Field Extensions and Towers: The problem is analyzed via a tower of field extensions:
$\mathbb{F}_p(t) \subset \mathbb{F}_p(x) \subset \mathbb{F}_p(x, h^2)$
- The extension $\mathbb{F}_p(x)/\mathbb{F}_p(t)$ is quadratic, generated by the square root of the discriminant of the defining quadratic equation for $x$ .
- The extension $\mathbb{F}_p(x, h^2)/\mathbb{F}_p(x)$ is an abelian extension of degree $(p-1)/2$ (related to the subgroup of squares $S$ ), derived from the fact that $h^2$ behaves like a root of a polynomial $H$ via Kummer theory.
Galois Group Analysis:
- The authors analyze the action of the non-trivial automorphism $\sigma$ of the quadratic extension $\mathbb{F}_p(x)/\mathbb{F}_p(t)$ on the polynomial $H$ (the truncation of $h$ ).
- They determine whether $\sigma$ can be extended to an involution (an automorphism of order 2) on the total extension $\mathbb{F}_p(x, h^2)$ .
- The existence of such an involution determines whether the total Galois group is cyclic or a product of cyclic groups, which in turn dictates whether the subfield generated by $f(t)$ has degree $p-1$ or $(p-1)/2$ over $\mathbb{F}_p(t)$ .
Congruence Conditions: The solvability of the involution condition depends on whether specific constants (derived from the rational parameterization) are quadratic residues modulo $p$ . This links the Galois group structure directly to congruence classes of $p$ (e.g., modulo 24, 12, or 8).

3. Key Contributions and Results

A. Apéry Numbers ( $f_\alpha$ )

The authors provide a rigorous proof for the pattern observed computationally in previous work.

Theorem 1: The Galois group $\text{Gal}(\mathbb{F}_p(t, f_\alpha)/\mathbb{F}_p(t))$ $Gal (F_{p} (t, f_{α}) / F_{p} (t))$ is:
- The subgroup of squares $S$ (index 2) if $p \equiv 1, 5, 7, 11 \pmod{24}$ .
- The full group $\mathbb{F}_p^\times$ if $p \equiv 13, 17, 19, 23 \pmod{24}$ .
Theorem 2: This corresponds to the factorization of the truncation polynomial $A_p(t)$ $A_{p} (t)$ :
- If $p \equiv 1, 5, 7, 11 \pmod{24}$ , $A_p(t)$ is a perfect square in $\mathbb{F}_p[t]$ .
- If $p \equiv 13, 17, 19, 23 \pmod{24}$ , $A_p(t) = (t^2 - 34t + 1) \cdot B_p(t)^2$ .

B. Domb Numbers ( $f_\delta$ ) and Almkvist–Zudilin Numbers ( $f_\xi$ )

The methodology is successfully adapted to these companion sequences.

Domb Numbers: The Galois group depends on $p \pmod 6$ $p (mod 6)$ .
- $p \equiv 1 \pmod 6 \implies$ Group is $S$ ; $A_{\delta, p}$ is a square.
- $p \equiv 5 \pmod 6 \implies$ Group is $\mathbb{F}_p^\times$ ; $A_{\delta, p}$ has a non-square factor $(64t^2 - 20t + 1)$ .
Almkvist–Zudilin Numbers: The Galois group depends on $p \pmod 8$ $p (mod 8)$ .
- $p \equiv 1, 3 \pmod 8 \implies$ Group is $S$ ; $A_{\xi, p}$ is a square.
- $p \equiv 5, 7 \pmod 8 \implies$ Group is $\mathbb{F}_p^\times$ ; $A_{\xi, p}$ has a non-square factor $(81t^2 + 14t + 1)$ .

C. Generalization and "Zoo" of Examples

The paper extends these findings to other Apéry-like sequences (Zagier's sporadic examples and sequences related to modular forms of specific levels).

Observation: For a wide class of sequences, the truncation $A_p(t)$ factors as $P(t) \cdot B_p(t)^2$ , where $P(t)$ is a polynomial determined by quadratic residue conditions on $p$ (specifically related to the level of the underlying modular form).
Tables 1 & 2: The authors provide computational evidence for numerous sequences (OEIS A229111, A290575, A274786, etc.), showing that the factorization pattern is governed by Legendre symbols (e.g., $\left(\frac{-5}{p}\right)$ , $\left(\frac{-7}{p}\right)$ ).

4. Significance

Rigorous Proof of Conjectures: The paper provides the first rigorous proofs for patterns regarding the Galois groups of Apéry-like series that were previously only observed computationally (e.g., in [CFV25]).
Uniformity of Galois Groups: It supports the conjecture that the Galois groups of reductions of D-finite series exhibit uniformity properties related to their differential Galois groups. The behavior is not random but strictly governed by congruence conditions on the prime $p$ .
Connection to Arithmetic Geometry: The work bridges the gap between the combinatorial properties of integer sequences (Lucas properties), the analytic properties of hypergeometric functions, and the arithmetic of field extensions.
Methodological Framework: The paper establishes a robust framework using rational parameterizations and Kummer theory that can be applied to other D-finite series arising from modular forms, suggesting a deep structural link between the level of the modular form and the splitting behavior of the associated generating series modulo primes.

In summary, the paper demonstrates that the "shadow" of the differential Galois group on the finite field level is determined by simple quadratic residue conditions, providing a complete classification for the most prominent Apéry-like sequences.