Correlations of zeros of a family of $L$-functions in function fields with symplectic symmetry

Technical Summary: Correlations of Zeros of a Family of L-Functions in Function Fields with Symplectic Symmetry

Problem Statement
The paper addresses the statistical distribution of zeros of $L$-functions within the function field setting, specifically focusing on the $n$-level density of zeros for families of quadratic Dirichlet $L$-functions. While the connection between $L$-function zeros and random matrix theory (RMT) is well-established for main terms, the precise calculation of lower-order terms and higher-level correlations ($n > 1$) remains a complex challenge. The authors aim to adapt the methodology developed by Mason and Snaith—originally formulated for number fields—to the context of function fields over $\mathbb{F}_q[T]$. The specific families under investigation are:

1. Quadratic Dirichlet $L$-functions associated with square-free monic polynomials of degree $2g+1$ (hyperelliptic curves of genus $g$).
2. $L$-functions associated with prime characters (monic irreducible polynomials of degree $2g+1$).

The goal is to derive explicit, conjectural formulas for the $n$-level density of zeros in these families, extending previous work that was largely limited to the 1-level density.

Methodology
The authors employ the "Ratios Conjecture" framework, adapted from the work of Conrey, Farmer, Zirnbauer, and Snaith, and specifically the set-theoretic reformulation introduced by Mason and Snaith. The core methodological steps include:

1. Set-Theoretic Reformulation: The authors rewrite the Ratios Conjecture for quadratic Dirichlet $L$-functions (originally proposed by Andrade and Keating) using set notation. They define sets of shifts $A = \{\alpha_1, \dots, \alpha_K\}$ and $B = \{\gamma_1, \dots, \gamma_Q\}$ and express the average of ratios of $L$-functions as a sum over subsets $D \subseteq A$. This formulation involves Euler products and zeta functions associated with the polynomial ring $\mathbb{F}_q[T]$, denoted as $\zeta_A(s)$.
2. Differentiation: To move from ratios to densities, the authors differentiate the conjectured ratio formula with respect to the shifts. By setting $B=A$ and differentiating, they derive an expression for the average of products of logarithmic derivatives of $L$-functions ($L'/L$). This process utilizes Lemma 5.1 (from Mason's thesis) to handle the differentiation of exponential sums over set partitions.
3. Residue Analysis: The paper analyzes the poles of the resulting expressions. Using the functional equation of the $L$-functions and the properties of $\zeta_A(s)$, the authors compute residues at specific poles (where shifts sum to zero). This leads to a recursive structure for the $n$-level density, allowing the reduction of $n$-point correlations to lower-order terms.
4. Contour Integration: The authors apply Cauchy's Residue Theorem to shift integration contours from the critical line to the imaginary axis. They demonstrate that the singularities of the integrand cancel out appropriately, allowing the derivation of a final integral formula for the $n$-level density.

Key Contributions and Results
The paper presents the following main results:

• Conjecture 3.1: A set-theoretic reformulation of the Ratios Conjecture for the family of quadratic Dirichlet $L$-functions over function fields. This provides a compact expression for the average of ratios of $L$-functions involving Euler products and specific zeta-function combinations.
• Theorem 3.2 & 3.3: Formulas for the differentiated ratios and the residues of the resulting expressions. These theorems establish the recursive relationships necessary to compute higher-order correlations.
• Theorem 3.4: The primary result, providing an explicit conjectural formula for the $n$-level density of zeros for the family of quadratic Dirichlet $L$-functions associated with hyperelliptic curves. The formula expresses the density as a sum over partitions of the set $\{1, \dots, n\}$, involving integrals of the differentiated ratio function $J^*_{DL}$ against a test function $f$.
• Theorem 3.5: An extension of the $n$-level density formula to the family of $L$-functions associated with prime characters. The authors note that the Euler product structure differs slightly from the quadratic case due to the nature of averaging over primes rather than all square-free polynomials.
• Verification (Section 8): The authors explicitly calculate the 1-level density using their general $n$-level formula. They demonstrate that this result perfectly matches the previously established 1-level density formula by Andrade and Keating (Theorem 1.2), thereby validating the consistency of their generalized approach.

Significance
The paper claims significance in the following areas:

• Generalization: It successfully generalizes the Mason and Snaith methodology to the function field setting, moving beyond the 1-level density to provide a framework for $n$-level densities.
• Consistency: By recovering the known 1-level density results for both quadratic and prime character families, the paper confirms that the adapted Ratios Conjecture framework is robust and consistent with existing rigorous results in the function field context.
• Arithmetic Insight: The derived formulas offer a systematic tool for predicting both leading and lower-order terms in the statistical behavior of zeros. This is crucial for understanding how subtle arithmetic features (such as the specific structure of the Euler products in function fields) influence the fine-scale distribution of zeros, potentially revealing deviations from the universal behavior predicted by Random Matrix Theory.

The work does not claim to prove the Ratios Conjecture but rather assumes its validity to derive explicit formulas for the $n$-level density, providing a concrete target for future rigorous verification in the function field context.