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Ties in function field prime race

This paper investigates ties in function field prime races by providing infinitely many examples of congruence classes that occur equally often for infinitely many $N$ , established through two distinct proofs utilizing explicit $L$ -function formulas with exceptional Galois conjugates and an explicit $\mathrm{GL}_2(\mathbb{F}_q)$ -action bijection.

Original authors: Graeme Bates, Ryan Jesubalan, Seewoo Lee, Jane Lu, Hyewon Shim

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Graeme Bates, Ryan Jesubalan, Seewoo Lee, Jane Lu, Hyewon Shim

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 are running a massive, infinite race. But instead of horses or runners, the competitors are polynomials (mathematical expressions like $T^2 + T + 1$ ) growing out of a finite field (a tiny, self-contained universe of numbers).

In this race, we are counting how many "prime" polynomials (the building blocks of all polynomials, much like prime numbers are for integers) appear in different "lanes." Each lane is defined by a specific remainder when divided by a fixed polynomial, let's call it the Modulus (the finish line marker).

The Big Question: Who is Winning?

For a long time, mathematicians have known that in these races, one lane often seems to have a slight advantage over another. This is called Chebyshev's Bias. It's like watching a marathon where, for the first 99% of the race, the runner in Lane 3 is always slightly ahead of the runner in Lane 1.

However, the paper you asked about isn't about who is winning. It's about ties.

The authors, a team of mathematicians, are asking: "Are there moments where the runners in different lanes are exactly neck-and-neck?"

They discovered that yes, these ties happen, but they are very specific. They don't happen randomly; they happen in predictable patterns based on the "length" of the race (the degree of the polynomials).

The Two Ways They Solved the Mystery

The paper presents two different "detective tools" to prove these ties exist. Think of them as two different ways to solve a puzzle.

1. The "Crystal Ball" Method (Explicit Formulas)

Imagine you have a magical crystal ball that can see the future of the race. In math, this is called an Explicit Formula involving $L$ -functions.

How it works: The authors look at the "vibrations" or "zeros" of these mathematical functions. Usually, these zeros are messy and unique, causing one lane to pull ahead.
The Twist: Sometimes, two zeros are twins (Galois conjugates). They are so perfectly matched that their effects cancel each other out. When this happens, the "vibrations" in Lane A and Lane B become identical, forcing the counts to be exactly the same.
The Analogy: It's like two musicians playing slightly different notes. Usually, the sound is a bit messy. But if they are playing perfect harmonics, the sound waves align perfectly, and the volume in both rooms becomes identical.

2. The "Magic Mirror" Method (GL2-Actions)

This is the more visual and creative approach.

How it works: Imagine the polynomials are dancers on a stage. The authors found a special "magic mirror" (a mathematical transformation called a $GL_2$ action) that can reflect the dancers.
The Trick: When you shine this mirror on a dancer in Lane A, they transform into a dancer in Lane B. Crucially, this mirror doesn't just swap them; it swaps every single dancer in Lane A with a unique partner in Lane B.
The Result: If you can pair every single runner in Lane A with a unique runner in Lane B, then the total number of runners in both lanes must be the same. It's a perfect one-to-one match.
The Catch: This mirror only works for specific race lengths (degrees). It's like a dance move that only looks good if the music is playing at exactly 120 beats per minute. If the music speeds up or slows down, the dance breaks, and the tie disappears.

Real-World Examples from the Paper

The authors tested this on several "universes" (finite fields):

The Binary Universe ( $F_2$ ): Here, the numbers are just 0 and 1. They found that for a specific modulus ( $T^3 + T + 1$ ), if the race length is 1 more than a multiple of 7, the lanes for $1$, $T$ , and $T+1$ are always tied. It's like a clock that resets every 7 laps, and at specific times, three runners are always side-by-side.
The Ternary Universe ( $F_3$ ): With numbers 0, 1, and 2, they found ties that depend on whether the race length is even or odd.

Why Does This Matter?

You might wonder, "Who cares if two lanes are tied?"

It breaks the bias: It proves that the "unfair advantage" one lane usually has isn't absolute. Nature finds a way to balance the scales, but only under very strict conditions.
It reveals hidden structure: Finding these ties is like finding a secret code in the universe of numbers. It tells us that the distribution of primes isn't random chaos; it has deep, hidden symmetries (like the "magic mirror" symmetry).
It challenges our intuition: The authors even conjecture that while these ties happen for specific race lengths, if you add up the total count from the start of the race to the current point, ties will almost never happen again. The "cumulative" race is much more chaotic than the "instantaneous" snapshot.

In a Nutshell

This paper is a detective story about the hidden order in the chaos of prime polynomials. The authors used two different detective kits—one looking at the "sound waves" of the numbers and the other using a "magic mirror" to swap them—to prove that sometimes, in the infinite race of primes, different lanes are perfectly tied, but only when the universe aligns just right.

1. Problem Statement

The paper investigates ties in prime number races within the context of function fields over finite fields, specifically $\mathbb{F}_q[T]$ .

Background: In classical number theory, Chebyshev's bias observes that primes congruent to $3 \pmod 4$ generally outnumber those congruent to $1 \pmod 4$ . Rubinstein and Sarnak formalized this using Dirichlet $L$ -functions, showing that while the bias exists, the difference $\Delta(x)$ changes sign infinitely often.
Function Field Analogue: Cha previously studied this bias in function fields, proving that under the Grand Simplicity Hypothesis (GSH) (the function field analogue of the Linear Independence conjecture), primes are biased toward non-quadratic residues. However, Cha also found examples where GSH fails, leading to unexpected biases.
The Specific Problem: This paper focuses on ties, defined as instances where the counts of irreducible monic polynomials in distinct congruence classes are exactly equal. Specifically, the authors seek pairs (or tuples) of distinct congruence classes $a, b \in (\mathbb{F}_q[T]/m)^\times$ such that:
$\pi(a, m, N) = \pi(b, m, N)$
holds for infinitely many degrees $N$ , often restricted to specific arithmetic progressions of $N$ .

2. Methodology

The authors employ two distinct and complementary approaches to prove the existence of these ties:

Approach A: Explicit Formulas and Galois Conjugacy (Analytic Method)

This method relies on the explicit formula for prime counting functions derived from Dirichlet $L$ -functions.

Matrix Möbius Inversion: The authors derive an exact formula for $\pi(a, m, N)$ using a matrix version of the Möbius inversion formula. They relate the counts to the zeros of $L$ -functions associated with Dirichlet characters modulo $m$ .
Galois Conjugates: The core mechanism for ties is the identification of "exceptional" Galois conjugate pairs among the zeros of the $L$ -polynomials. If the zeros (or their traces) corresponding to different congruence classes are Galois conjugates, their contributions to the explicit formula cancel out or become identical for specific $N$ .
Conditions: This method identifies ties based on the arithmetic properties of the degree $N$ relative to the modulus $m$ and the structure of the cyclotomic fields generated by the $L$ -function zeros.

Approach B: Explicit Bijections via Group Actions (Algebraic/Geometric Method)

This method constructs a direct combinatorial bijection between sets of irreducible polynomials in different congruence classes.

GL $_2(\mathbb{F}_q)$ Action: The authors define an action of the group $GL_2(\mathbb{F}_q)$ on polynomials $f(T)$ of degree $N$ :
$(f|_N B)(T) = (\gamma T + \delta)^N f\left(\frac{\alpha T + \beta}{\gamma T + \delta}\right)$
where $B = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \in GL_2(\mathbb{F}_q)$ .
Preservation of Properties: They prove that for specific matrices $B$ and moduli $m$ (where $m$ is fixed or scaled by $B$ ), this map preserves irreducibility and degree.
Monicity and Ties: While the map generally changes the leading coefficient, the authors identify specific cases (e.g., characteristic 2, or specific choices of $B$ ) where the map preserves monicity. This establishes a bijection between the set of monic irreducibles in class $c$ and class $c|_N B$ , proving $\pi(c, m, N) = \pi(c|_N B, m, N)$ .
Periodicity: The congruence class of the image depends on $N$ modulo the order of a specific element in $(\mathbb{F}_q[T]/m)^\times$ , leading to ties that hold for $N$ in specific arithmetic progressions.

3. Key Contributions and Results

General Theoretical Framework

Exact Counting Formula: The paper provides a rigorous derivation of $\pi(a, m, N)$ using matrix Möbius inversion (Proposition 3.1), allowing for exact computation without error term bounding.
Two-Pronged Proof Strategy: The paper demonstrates that analytic properties (Galois conjugacy of $L$ -function zeros) and algebraic symmetries ( $GL_2$ actions) yield the same results, offering a deeper understanding of the underlying structure of function field primes.

Specific Examples of Ties

The authors provide infinite families of examples where ties occur, including cases in characteristic 2 (previously less explored in this context):

Modulus $m = T^3 + T + 1 \in \mathbb{F}_2[T]$ :
- For $N \equiv 1 \pmod 7$ , the authors prove:
  $\pi(1, m, N) = \pi(T, m, N) = \pi(T+1, m, N)$
  $\pi(T^2, m, N) = \pi(T^2+T, m, N) = \pi(T^2+T+1, m, N)$
- This is explained by the fact that the zeros of the relevant $L$ -functions are Galois conjugates, and also by the action of a specific matrix in $GL_2(\mathbb{F}_2)$ of order 7.
Modulus $m = T^2 + T + 1 \in \mathbb{F}_2[T]$ :
- Ties occur based on $N \pmod 3$ . For instance, if $N \equiv 1 \pmod 3$ , then $\pi(1, m, N) = \pi(T, m, N)$ .
Modulus $m = T^2 + 1 \in \mathbb{F}_3[T]$ :
- Ties are established for odd and even $N$ separately, grouping congruence classes based on the action of diagonal matrices in $GL_2(\mathbb{F}_3)$ .
Modulus $m = T^2 \in \mathbb{F}_p[T]$ :
- For $p=2$ , ties exist between $1$ and $T+1$ for odd $N$ .
- For odd $p$ , ties exist between classes related by multiplication by generators of $\mathbb{F}_p^\times$ .
Artin-Schreier Polynomials:
- For $m = T^p - T - 1$ , translation symmetries induce ties between classes $c(T)$ and $c(T+b)$ .

Bias Analysis

The paper also analyzes cases where ties do not occur, revealing consistent biases. For example, in the case $m = T^2 + T + 1 \in \mathbb{F}_2[T]$ with $N$ being a multiple of 3, there is a consistent bias against the class $1 $(i.e.,$ \pi(1, m, N) < \pi(T, m, N)$ for large $N$ ).

4. Significance and Future Directions

Unification of Methods: The paper successfully bridges analytic number theory (via $L$ -functions and Galois theory) and algebraic geometry (via group actions on polynomial spaces). It shows that "exceptional" Galois conjugates in the analytic approach correspond precisely to symmetries in the geometric approach.
Characteristic 2: The results significantly expand the understanding of prime races in characteristic 2, a case where previous literature was sparse due to the lack of quadratic residue distinctions in the same way as odd characteristics.
Conjecture on Cumulative Ties: The authors conjecture that while pointwise ties (for a specific degree $N$ ) are common and structured, cumulative ties (where the sum of counts up to $N$ are equal) are extremely rare or non-existent for large $N$ . They provide numerical evidence supporting this, suggesting that the "race" rarely results in a tie when accumulated over time.
Computational Tools: The authors provide SageMath code to generate these tables and verify the explicit formulas, facilitating further research in this domain.

In summary, this paper provides a comprehensive classification of when and why prime counts in function fields can be exactly equal, utilizing a powerful combination of explicit analytic formulas and geometric group actions.