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p-Adically convergent loci in varieties arising from periodic continued fractions

Inspired by alternative definitions of $p$ -adic continued fractions, this paper investigates the $p$ -adically convergent loci of algebraic varieties representing periodic continued fractions with partial quotients in $\mathbb{Z}[1/p]$ that satisfy a quadratic equation, specifically characterizing their zero and one-dimensional cases.

Original authors: Laura Capuano, Marzio Mula, Lea Terracini, Francesco Veneziano

Published 2026-07-13

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CC BY 4.0

Original authors: Laura Capuano, Marzio Mula, Lea Terracini, Francesco Veneziano

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 magical number machine that takes a list of numbers and spits out a single, infinite fraction. In the world of normal math (the "Archimedean" world), we know exactly how this machine works: if you feed it a list that repeats itself over and over, it usually settles down to a specific number, like a pendulum swinging until it stops. This is the classic story of continued fractions, a tale as old as Lagrange.

But what happens if we swap our normal ruler for a strange, p-adic ruler? In this p-adic world, numbers behave like they are made of a different kind of clay. The rules of closeness are flipped: two numbers are "close" if their difference is divisible by a huge power of a prime number $p$ . In this weird universe, the same infinite fraction machine might spin wildly forever, never settling on a single answer.

This paper is a detective story about finding the specific "addresses" (mathematical varieties) where these p-adic fraction machines do actually stop spinning and converge to a real number. The authors, Laura Capuano and her team, are exploring a landscape of algebraic shapes (varieties) where each point represents a repeating fraction. Their mission? To map out exactly which of these points are "convergent" and which are chaotic dead ends.

The Golden Rule of Convergence

The team discovered a simple, strict rule (Theorem 3.1) that acts like a traffic light for these fractions. For a repeating fraction to settle down in the p-adic world, two things must happen:

The Sum Must Be Big: A specific combination of the repeating numbers must be "large" in the p-adic sense (meaning it's divisible by a high power of $p$ ).
No Zero Divisions: A series of checks on the intermediate steps must ensure we don't accidentally divide by zero or hit a wall.

If these conditions aren't met, the fraction is a lost cause; it will never converge. The authors prove this isn't just a guess—it's a hard, mathematical fact.

Mapping the Landscape: Small Dimensions

The authors zoom in on the simplest shapes in this landscape, where the repeating patterns are short (lengths of 1, 2, or 3). They act like cartographers, drawing the boundaries of where convergence is possible.

The One-Step Loop (Type 0,1): Here, the fraction just repeats one number. The paper proves that for this to work, the number must be a very specific rational value. If the polynomial defining the loop has no roots in the p-adic world, the map is empty. There are no irrational square roots hiding here; the machine simply won't converge to them.
The Two-Step Loop (Type 0,2): When the pattern repeats two numbers, the authors find that convergence is still extremely rare. They show that if it does happen, the product of those two numbers must be a very specific type of square. Again, no irrational square roots can be the destination here.
The Three-Step Loop (Type 0,3): This is where things get spicy. When the pattern has three numbers, the landscape opens up, but it's still a desert. The authors prove that for a specific type of number (pure radicals like $\sqrt{d}$ ), the number of convergent fractions is finite. They don't just say "it's rare"; they prove there are only a limited number of solutions, like finding a handful of specific keys that fit a complex lock.

The Pell Equation Connection

For the three-step loops involving square roots (like $\sqrt{d}$ ), the authors uncover a hidden link to Pell equations. These are ancient Diophantine puzzles (equations like $x^2 - dy^2 = n$ ) that have been studied for centuries.

The paper shows that finding a convergent p-adic fraction for $\sqrt{d}$ is exactly the same as finding a solution to a specific Pell equation. This is a powerful tool because we know how to generate families of solutions to Pell equations using a "fundamental unit" (a master key). However, the authors use this to prove a negative result: even though there are infinite families of solutions to the Pell equation, only a finite number of them will satisfy the strict p-adic convergence rules.

They even construct specific examples. For instance, they show how to build convergent fractions for $\sqrt{10}$ or $\sqrt{5}$ using specific prime numbers, but they are careful to note that these are isolated islands of success, not a vast continent.

What They Rule Out

It is crucial to understand what this paper says doesn't work.

No Magic Algorithms: The authors explicitly state they are not looking at a fixed algorithm (like the famous Browkin or Ruban methods) that generates fractions for any number. Instead, they look at any repeating sequence.
No Irrational Square Roots in Simple Loops: For the simplest loops (lengths 1 and 2), the paper proves it is impossible to converge to an irrational square root. If you try to force a simple repeating fraction to equal $\sqrt{2}$ or $\sqrt{3}$ in the p-adic world, it simply won't happen.
No Infinite Solutions for Simple Cases: While some complex cases might have infinite families, the authors prove that for the "pure radical" cases (like $\sqrt{d}$ with a 3-step loop), the number of solutions is finite.

The Verdict: Finite and Precise

The authors are not merely suggesting that these convergent points exist; they have proved the finiteness of the solution sets for the cases they studied. They didn't just run a computer simulation and say, "It looks like there are a few." They used deep tools from algebraic geometry and number theory to show that the set of convergent points is a finite collection of specific coordinates.

For example, in the case of $\sqrt{10}$ with a 3-step loop, they explicitly list the few specific fractions that work, such as $[13, 9/53, -4]$ . They also show that for certain types of numbers (like those divisible by 4 or primes congruent to -1 mod 4), the set of solutions is empty—there are no convergent fractions at all.

The Big Picture

This paper is a rigorous map of a tiny, tricky corner of the mathematical universe. It tells us that while the p-adic world is full of repeating patterns, the ones that actually settle down to a number are incredibly picky. They require a perfect alignment of algebraic conditions. The authors have drawn the borders, proved that the borders are finite, and shown us exactly where the "safe zones" are. They haven't found a way to make every number converge, but they have successfully identified the rare, special cases where the p-adic machine finally stops spinning and gives us a clear answer.

Technical Summary: p-ADICALLY CONVERGENT LOCI IN VARIETIES ARISING FROM PERIODIC CONTINUED FRACTIONS

Problem Statement
The paper investigates the $p$ -adic convergence of periodic continued fractions (PCFs) with partial quotients in the ring of $S$ -integers $\mathbb{Z}[1/p]$ , where $p$ is an odd prime. While the classical (Archimedean) theory of continued fractions is well-established—linking periodicity to quadratic irrationals via Lagrange's theorem—the $p$ -adic setting presents distinct challenges. Unlike the classical case, there is no single canonical algorithm for generating $p$ -adic continued fractions (e.g., Ruban, Browkin, Schneider algorithms exist, but they are not unique). Consequently, the authors do not restrict their study to expansions generated by a specific algorithm. Instead, they adopt a geometric perspective inspired by Brock, Elkies, and Jordan (BEJ21), who defined algebraic varieties whose points represent formal periodic continued fractions satisfying a given quadratic equation. The central problem is to characterize the $p$ -adically convergent locus of these varieties: the subset of points corresponding to continued fractions that converge with respect to the $p$ -adic absolute value rather than the Archimedean one.

Methodology
The authors employ a synthesis of algebraic geometry, arithmetic, and the theory of linear recurrences and Pell equations.

Geometric Framework: They utilize the varieties $V(F)_{N,k}$ , where $F(x) = Ax^2 + Bx + C$ is a quadratic polynomial, and $(N, k)$ denotes the lengths of the pre-period and period, respectively. Points in these varieties correspond to tuples $(b_1, \dots, b_N, a_1, \dots, a_k)$ representing the PCF $[b_1, \dots, b_N, \overline{a_1, \dots, a_k}]$ .
Convergence Criterion: A pivotal tool is Theorem 3.1, which provides a necessary and sufficient criterion for the $p$ -adic convergence of a purely periodic continued fraction $P = [a_1, \dots, a_k]$ . The criterion relies on the eigenvalues of the matrix $M(a_1, \dots, a_k)$ associated with the period. Specifically, convergence requires that the trace of this matrix has $p$ -adic absolute value greater than 1, and that specific entries of the matrix powers satisfy strict $p$ -adic valuation inequalities.
Case-by-Case Analysis: The authors analyze varieties of small dimension (where $N+k \leq 3$ $N + k \leq 3$ ). For each type $(N, k)$ $(N, k)$ , they:
- Characterize the set of $\mathbb{Z}[1/p]$ -rational points on $V(F)_{N,k}$ .
- Apply Theorem 3.1 to filter these points for $p$ -adic convergence.
- Express the convergent points explicitly in terms of the coefficients of $F$ where possible.
Diophantine Connections: For the case of "pure radicals" ( $F(x) = x^2 - d$ ) with type $(0, 3)$ , the defining equations of the variety reduce to generalized Pell equations. The authors leverage the theory of linear recurrence sequences and the unit group action on Pell solutions to derive finiteness results and construct explicit families of convergent expansions.

Key Contributions and Results

General Convergence Criterion: The paper establishes Theorem 3.1, a sharp criterion for $p$ -adic convergence of periodic fractions. This result is shown to be distinct from previous sufficient criteria (e.g., by Browkin) and highlights that convergence depends on the periodic part's matrix properties, not just the pre-period.
Explicit Enumeration for Small Dimensions:
- Types (0,1), (1,1), (0,2): The authors provide complete characterizations of the convergent loci. For instance, for type (0,1), convergence occurs if and only if the unique rational point satisfies a specific $p$ -adic inequality involving the coefficients of $F$ .
- Types (2,1), (0,3): The authors prove the finiteness of the $p$ -adically convergent locus for these types. For type (0,3) with $F(x) = x^2 - d$ , the set of convergent points is finite for any fixed $d$ .
- Type (1,2): The convergent locus is shown to be finite, and the paper provides conditions under which it is non-empty, relating the existence of solutions to the reducibility of $F$ and specific $p$ -adic valuations.
Construction of Explicit Families:
- For $F(x) = x^2 - d$ with $d = a^2 + 1$ , the authors construct infinite families of $p$ -adically convergent PCFs of type (0,3) by identifying prime powers within specific linear recurrence sequences (related to Chebyshev polynomials).
- They construct explicit convergent periodic expansions for square roots of infinitely many integers, including forms like $\sqrt{p^2+1}$ and $\sqrt{a^2+1}$ .
- In the case of type (1,3), they provide explicit families of convergent expansions for quadratic irrationals, including examples like $\sqrt{10}$ and $\sqrt{p^2+1}$ .
Non-Existence Results: The paper demonstrates that for certain types (e.g., (0,1) and (0,2)), irrational square roots cannot be limits of convergent PCFs, consistent with known results on Browkin continued fractions. It also identifies conditions (e.g., $d \leq 0$ or specific modular constraints) under which no convergent PCFs of type (0,3) exist for $\sqrt{d}$ .

Significance and Claims
The paper claims to provide the first systematic geometric study of $p$ -adic convergence for periodic continued fractions without relying on a fixed generation algorithm. By shifting the focus from algorithmic generation to the geometric properties of the varieties $V(F)_{N,k}$ , the authors are able to:

Unify the treatment of various $p$ -adic expansions under a single algebraic framework.
Derive effective finiteness results for the number of convergent expansions for a given quadratic irrational in low-dimensional cases.
Construct explicit, non-trivial families of convergent $p$ -adic expansions for square roots of integers, demonstrating that such expansions exist for infinitely many integers and primes, even when they do not arise from standard "floor function" algorithms.

The authors note that while their results for small dimensions are explicit, the relationship between convergent points and the polynomial $F$ becomes less transparent for higher dimensions. They emphasize that their work connects the theory of $p$ -adic continued fractions with deep arithmetic questions regarding Pell equations and linear recurrence sequences, offering a new perspective on the convergence of these formal series.