Bounded Level Primes Break Density Universality at First Order

The Katz–Sarnak conjecture predicts that low-lying zeros of $L$-functions settle into a universal distribution determined by symmetry type. That limit is deliberately coarse: it discards how a family was assembled once its analytic conductor becomes large. This paper asks whether the first finite-conductor correction retains that arithmetic memory for holomorphic cusp newforms of fixed weight and varying level.

The authors show that it does. For harmonically weighted one- and two-level densities, the first deviation from an explicit profile depending only on the weight, test functions, and conductor is governed by a single family-dependent quantity, $\kappa_1(F)$. Its dependence on the bounded prime divisors of the level gives two otherwise comparable families a detectable separation at order $1/\log R$.

Core Contribution

The central result is a precise failure of universality in the lower-order terms, not in the Katz–Sarnak limit itself. Families with the same weight, the same symmetry type, and conductors of comparable size still share the limiting density. But the paper proves that their first corrections need not agree when their levels have different bounded prime factors.

This distinction matters because lower-order terms are where finite-conductor data enter density predictions. Treating symmetry type as the whole story is correct only at the limiting scale. The authors identify the extra arithmetic datum explicitly rather than merely showing that an error term exists.

Technical Approach

The analysis considers families of holomorphic cusp newforms with fixed weight $k$ and varying level $N$, ordered through analytic conductor $R$. The authors work with harmonic weights and sharpen earlier one-level-density work of Miller. Their formulas separate a universal component—built from $k$, the chosen test functions, and $R$—from a residual term that factors into a test-function functional and $\kappa_1(F)$.

That factorization is the useful structural point. The test functions determine which spectral statistic is being probed, while $\kappa_1(F)$ carries the family-specific contribution from bounded primes dividing the level. The arithmetic effect is therefore not presented as an unspecified correction hidden in asymptotic notation; it is isolated as a computable invariant.

For the one-level density, the stated remainder is $O(\log^{-5}(R))$. For the two-level density, it is $O(\log^{-2}(R))$. Since the nonuniversal contribution occurs at order $1/\log R$, both estimates leave room to identify that leading correction rather than confounding it with the remainder.

Results and Analysis

The paper establishes the first lower-order term exactly for both statistics. In the one-level case, the $O(\log^{-5}(R))$ control lies four logarithmic orders below the displayed $1/\log R$ effect. In the two-level case, the $O(\log^{-2}(R))$ remainder is one order below it. This is sufficient asymptotic separation to support the claimed comparison between families.

The result changes the interpretation of finite-conductor density calculations. A match to the Katz–Sarnak prediction cannot by itself establish that two families behave identically at the scale visible in lower-order expansions. The paper supplies a mechanism for the discrepancy: different bounded prime factors in the level alter $\kappa_1(F)$, and hence alter the coefficient multiplying the test-function functional.

The evidence is analytic rather than experimental. That is appropriate for the claim: the contribution is an asymptotic formula with explicit error bounds, not an empirical pattern inferred from sampled zeros. Within the harmonic, fixed-weight newform setting studied here, symmetry type does not determine the first correction.

Scope and Caveats

The result concerns holomorphic cusp newforms at fixed weight with varying level and harmonic weighting. It does not by itself establish the same correction structure for unweighted families, variable weights, or other automorphic settings. It also resolves the first lower-order term rather than a full hierarchy of finite-conductor corrections.

For practitioners using density conjectures as a model of finite families, the practical lesson is to retain local level data when comparing lower-order statistics. The limiting symmetry class remains the right leading description; $\kappa_1(F)$ explains why that description is incomplete one logarithmic order later.