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The first moment of quadratic Dirichlet $L$ -functions in the even hyperelliptic ensemble

This paper establishes an asymptotic formula for the first moment of central values of quadratic Dirichlet $L$ -functions in the even hyperelliptic ensemble over $\mathbb{F}_q[x]$ , revealing a novel secondary term with coefficients that exhibit a period-three dependence on the genus $g$ modulo 3, a feature distinct from previous even-degree results.

Original authors: Hwanyup Jung

Published 2026-09-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Hwanyup Jung

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

In the vast landscape of mathematics, there is a branch dedicated to understanding the hidden rhythms of numbers. At the heart of this field lie objects called L-functions, which are complex formulas that encode deep information about how numbers are distributed. Mathematicians often study these formulas by looking at their values at a specific central point, much like checking the temperature at the exact center of a storm to understand its intensity. When these formulas are linked to quadratic characters—mathematical tools that distinguish between different types of numbers based on their remainders—they form a family known as quadratic Dirichlet L-functions. A major goal in this area is to calculate the average value of these central points across a large collection of related formulas. This average, or "first moment," helps researchers test grand theories about the behavior of prime numbers and the structure of number systems. While the behavior of these averages has been well understood for certain types of number systems, a specific and stubborn case involving even-degree polynomials over finite fields had resisted a complete description, leaving a gap in the theoretical map.

A researcher named Hwanyup Jung has now filled that gap with a precise and rigorous calculation. The study focuses on a specific collection of polynomials over a finite field, which can be thought of as a number system with a limited, fixed number of elements. In this setting, the polynomials are arranged in a family called the even hyperelliptic ensemble. The challenge was to find an exact formula for the average of the central values of the L-functions attached to these polynomials as the size of the polynomials grows infinitely large. Previous attempts had successfully identified the main trend of this average, but they missed a subtle, secondary pattern that appeared to depend on the specific size of the polynomials in a way that earlier theories did not predict.

Jung's breakthrough came from a change in perspective. Instead of working directly with the standard L-function, the researcher worked with a "completed" version of it. This modified function is mathematically cleaner and possesses a perfect symmetry that the original lacks. By using this symmetric version, the researcher was able to apply a powerful technique called Poisson summation, which acts like a lens to reveal hidden structures in the data. This approach allowed for a precise calculation of the average value, revealing that the result is not just a single smooth curve. Instead, the formula contains a secondary term that oscillates in a very specific way.

The most striking discovery is that this secondary term does not behave the same way for every size of polynomial. Its value depends on the size of the polynomial modulo three. In simpler terms, if you count the size of the polynomials and divide by three, the remainder—whether it is zero, one, or two—determines the exact shape of this secondary contribution. This creates a repeating pattern of three distinct behaviors. The researcher proved that the coefficients governing this pattern are real numbers and that the cycle of three is the shortest possible period; the pattern does not repeat more frequently than this. This finding contradicts earlier formulas for similar problems, which suggested that such a dependence on the remainder modulo three should not exist.

To ensure this result was not just a theoretical artifact, the study included a numerical verification. By computing the exact averages for small, manageable sizes of polynomials and comparing them to the new formula, the researcher confirmed that the predicted pattern matched the actual data perfectly. The numbers showed that the average value indeed swings between three different behaviors depending on the size of the polynomial, just as the new formula predicted. The error in the approximation was also tightly controlled, shrinking rapidly as the polynomials grew larger, which gives high confidence in the accuracy of the result.

This work resolves a long-standing question about the first moment of these L-functions in the even-degree case. It demonstrates that the behavior of these mathematical objects is more intricate than previously thought, carrying a subtle three-fold rhythm that was invisible to earlier methods. By establishing an exact asymptotic formula that captures both the main trend and this oscillating secondary term, the study provides a complete picture of how these averages behave. It stands as a definitive proof that the coefficients of the secondary term are real and that their minimal period is exactly three, offering a clearer and more accurate understanding of the underlying structure of these number systems.

Problem Statement
This paper investigates the first moment of central values of quadratic Dirichlet $L$ -functions over the even hyperelliptic ensemble $\mathcal{H}_{2g+2}$ in the function field $\mathbb{F}_q[x]$ , where $q$ is a fixed odd prime power. Specifically, the author seeks an asymptotic formula for the sum:
$M(g) := \sum_{D \in \mathcal{H}_{2g+2}} L\left(\frac{1}{2}, \chi_D\right)$
as the genus $g \to \infty$ . While the principal term and error exponent for this problem were previously established by Jung [10] and Andrade–MacMillan [3], those results contained lower-order terms of order $g q^{2g/3}$ whose coefficients were independent of $g \pmod 3$ . Furthermore, previous works often imposed congruence conditions on $q$ (e.g., $q \equiv 1 \pmod 4$ ). This paper aims to remove these restrictions and refine the secondary term analysis to reveal a previously unobserved periodic dependence on $g$ .

Methodology
The analysis departs from earlier treatments in two primary ways:

Use of the Completed $L$ -function: Instead of working directly with $L(s, \chi_D)$ , the author works with the completed $L$ -function $L^*(s, \chi_D) = L(s, \chi_D)(1-q^{-s})^{-1}$ . For the even ensemble, the character $\chi_D$ is even, causing $L(s, \chi_D)$ to carry a trivial factor $(1-q^{-s})$ . Removing this factor yields $L^*(s, \chi_D)$ , which satisfies a symmetric functional equation and admits an exact two-term central-value formula involving truncation levels $g$ and $g-1$ . This approach avoids the induction on $g$ and residue calculations at the origin required in previous even-degree treatments.
Phase-Normalized Gauss Sums: To handle arbitrary odd prime powers $q$ without the restriction $q \equiv 1 \pmod 4$ , the author introduces a phase-normalized Gauss sum. Standard quadratic Gauss sums over $\mathbb{F}_q[x]$ carry a root-of-unity phase that depends on $q \pmod 4$ . By absorbing this phase into the definition of the Gauss sum, the resulting Poisson summation formula becomes phase-free for all odd $q$ . This allows the application of techniques previously limited to the $q \equiv 1 \pmod 4$ case.

The core of the proof involves decomposing the moment $M^*(g)$ (the sum of completed $L$ -values) into three contributions based on the dual variables arising from Poisson summation:

Zero dual variable: Yields the principal term.
Square dual variables: Yields the secondary term.
Non-square dual variables: Bounded as an error term.

A critical technical step involves shifting contours in the complex plane. The author retains the contour integral from the zero-dual contribution and shows it is exactly cancelled by a corresponding contour integral arising from the principal square-dual residues. The remaining secondary term arises from the residues at the "cubic points" $z^3 = q^{-4}$ .

Key Contributions and Results

Exact Asymptotic Formula: The paper establishes the following asymptotic formula for $M(g)$ :
$M(g) = M_{\text{prin}}(g) + S(g) + O_\epsilon(q^{g(1+\epsilon)/2})$
where $M_{\text{prin}}(g)$ is the principal term (matching previous results) and the secondary term is:
$S(g) = q^{2g/3+2} (a_g g + b_g)$
Period-Three Coefficients: The coefficients $a_g$ and $b_g$ are real numbers that depend on $g \pmod 3$ . The sequence $m \mapsto (a_m, b_m)$ has a minimal period of exactly three. This contrasts with earlier even-degree formulas where the secondary coefficients were constant with respect to $g \pmod 3$ . The periodicity arises because the secondary term is a sum of residues at the three cube roots of $q^{-4}$ (i.e., $z_k = q^{-4/3} e^{2\pi i k/3}$ for $k=0,1,2$ ), rather than just the real root.
Generalization to Arbitrary Odd $q$ : The results hold for any fixed odd prime power $q$ , removing the need for congruence conditions like $q \equiv 1 \pmod 4$ that were present in prior literature.
Numerical Verification: The paper provides numerical data for small $q$ and $g$ confirming that the sign pattern of the secondary term oscillates with period three, matching the theoretical prediction. The data also rules out alternative models (such as period-two oscillations) for the secondary term.

Significance
The paper claims that the primary significance of this work lies in the discovery of the period-three behavior in the secondary term of the first moment for the even hyperelliptic ensemble. This behavior was previously obscured because earlier methods retained only the real residue, effectively averaging out the complex phases. By utilizing the completed $L$ -function and a phase-normalized Poisson summation, the author reveals that the lower-order terms are sensitive to $g \pmod 3$ .

The author notes that the method of working with the completed $L$ -function and the exact two-term formula is expected to extend to higher moments ( $k \ge 2$ ) over the even ensemble, potentially simplifying the combinatorial complexity (reducing $4^k$ blocks to $2^k$ ) compared to previous approaches. However, the paper focuses strictly on the first moment and does not provide explicit results for higher moments.

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