One-level density of zeros of $\Gamma_1(q)$ $L$-functions

Technical Summary: One-Level Density of Zeros of $\Gamma_1(q)$ L-Functions

Problem Statement
The paper investigates the statistical distribution of zeros of automorphic $L$-functions, specifically focusing on the one-level density of a large unitary family of $GL_2$ $L$-functions associated with the congruence subgroup $\Gamma_1(q)$. The central problem is to determine the admissible support of the Fourier transform of the test function, $\hat{\phi}$, for which the one-level density matches the prediction of the Katz–Sarnak philosophy.

While the Katz–Sarnak conjecture predicts that the local statistics of zeros in natural families correspond to eigenvalues of random matrices from a specific symmetry group (in this case, the unitary group $U$), previous results for unitary families have been restricted to a support of $\text{supp } \hat{\phi} \subset (-2, 2)$. Extending this support beyond the "diagonal" range is a significant challenge in analytic number theory, as it requires controlling off-diagonal terms in the explicit formula with greater precision. The paper seeks to answer whether the structural properties of the $L$-function family (specifically the $GL_2$ structure) allow for a larger support than what is dictated solely by the symmetry group.

Methodology
The author employs the explicit formula to relate the sum of the test function over the zeros of the $L$-functions to sums over prime powers. The core technical work involves bounding the off-diagonal contributions arising from the Petersson trace formula applied to the family of cusp forms in $S_k(\Gamma_1(q))$.

Key methodological steps include:

1. Explicit Formula and Trace Formula: The one-level density is expressed as an integral plus a sum over prime powers involving Fourier coefficients $\lambda_f(n)$. The sum over the family is evaluated using a Petersson-type formula, which decomposes the sum into a diagonal term (yielding the main term) and off-diagonal terms involving Bessel functions and exponential sums.
2. Decomposition of Off-Diagonal Terms: The off-diagonal sum is split based on the relationship between the prime $p$ and the moduli $s, t$ arising from the trace formula. The sum is further decomposed into contributions from the trivial character and non-trivial Dirichlet characters.
3. Dyadic Decomposition and Mellin Transforms: The sums are analyzed in three distinct transition regions defined by the behavior of the Bessel function $J_{k-1}$ and the exponential terms:
   • Case 1: $P/q \ll ST$ (where $P$ is the size of the prime and $S, T$ are sizes of $s, t$).
   • Case 2: $\sqrt{P}/q \ll ST \ll P/q$.
   • Case 3: $\sqrt{P}/q \gg ST$.
     In these regions, the author utilizes dyadic decomposition, Mellin transform representations of the exponential and Bessel functions, and contour shifting techniques.
4. Assumption of GRH: The Generalized Riemann Hypothesis (GRH) is assumed throughout to bound the $L$-function derivatives and to shift integration contours to the right, ensuring the convergence and smallness of error terms.
5. Character Sums: For non-trivial characters, the author applies Cauchy-Schwarz inequalities and orthogonality relations of Dirichlet characters to bound the sums, leveraging Lemma 2.4 (bounds for prime sums twisted by characters) to control the error terms.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes the one-level density for the family of $\Gamma_1(q)$ $L$-functions (with weight $k \ge 3$ odd and $q$ prime) under GRH.

• Extended Support: The paper proves that the Katz–Sarnak prediction holds for test functions with Fourier transform support in the interval $\text{supp } \hat{\phi} \subset (-8/3, 8/3)$. This extends the previously known bound of $(-2, 2)$ for unitary families.
• Non-Vanishing Proportion: As a direct application (Corollary 1.2), the author derives a lower bound for the proportion of forms in the family that do not vanish at the central point $s=1/2$. The result states that at least $5/8$ (62.5%) of the forms in the family are non-vanishing at the central point.
• Structural Insight: The paper demonstrates that the $GL_2$ structure of the family is the decisive factor in extending the support, rather than just the unitary symmetry group. This challenges the notion that the symmetry group alone dictates the bandwidth limit, showing that larger families with specific structural properties (like the $GL_2$ family here) can achieve supports comparable to or exceeding those of orthogonal families.

Significance
The paper claims that the 62.5% non-vanishing proportion is the highest record for any family associated with a unitary group. The significance of the work lies in two main areas:

1. Refined Arithmetic Information: By extending the support of the test function to $(-8/3, 8/3)$, the result provides more refined arithmetic information, specifically stronger bounds on the proportion of non-vanishing central $L$-values.
2. Structural Understanding: The work indicates that the "bandwidth limit" for the Fourier transform support is not solely determined by the symmetry group (unitary vs. orthogonal) but is heavily influenced by the finer structural properties of the family of $L$-functions. The ability to push the support to $8/3$ for a unitary $GL_2$ family, whereas previous unitary $GL_1$ results were limited to $2$ (or slightly above with extra hypotheses), highlights the power of the $GL_2$ structure in controlling off-diagonal terms.

The author notes that the techniques developed can be extended to study higher-level densities of the same family with the same support, a direction currently being investigated. The paper does not propose new experimental applications but focuses on the theoretical advancement of the one-level density conjecture within the framework of analytic number theory.