The Dimension Split in L-Function Zero Statistics: When GUE Meets Poisson
Number TheoryL-FunctionsRandom Matrix TheoryRiemann HypothesisSpectral StatisticsGUEMontgomery-OdlyzkoLMFDB
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In 1973, Hugh Montgomery proposed a startling conjecture: the non-trivial zeros of the Riemann zeta function are distributed like the eigenvalues of random Hermitian matrices from the Gaussian Unitary Ensemble (GUE). 14 years and 175 million zeros later, Andrew Odlyzko's numerical computations at height ~10²⁰ confirmed it — the spacing histogram and GUE's prediction are essentially indistinguishable.
The Montgomery–Odlyzko law became folklore: all L-function zeros should follow GUE statistics. The Katz–Sarnak philosophy (1999) formalized this by symmetry type, but the core intuition remained: zeros repel each other quadratically, like charged particles on a line.
But is this universal? Does every L-function, regardless of its arithmetic complexity, converge to the same random-matrix limit?
We tested this on 63,844 weight-2 newforms from the LMFDB — the largest such analysis to date — and found something that surprised us.
For each of 63,844 modular forms, we took the 10 lowest non-trivial L-function zeros (568,708 nearest-neighbor spacings in total). We unfolded them (dividing by the mean spacing per form) and fitted the Brody distribution — a one-parameter family that interpolates between Poisson (β=0, no repulsion) and GUE (β=2, quadratic repulsion):
P(s; β) = (β+1) · a · s^β · exp(−a · s^(β+1))
The parameter β quantifies level repulsion. β=0 means zeros behave like independent random points (a Poisson process). β=2 means they repel like GUE eigenvalues.
We then stratified the analysis by Hecke field dimension (the degree of the number field generated by the Fourier coefficients of the form).
| Group | β (MLE) | Interpretation |
|---|---|---|
| All forms (aggregate) | 0.620 | "Intermediate repulsion" — misleading |
| dim = 1 (CM forms) | 1.879 | GUE-like (β=2) |
| dim ≥ 2 (non-CM) | 0.242 | Near-Poisson (β=0) |
| Rank 0 | 0.676 | Intermediate |
| Rank 1 | 0.538 | Intermediate |
One-dimensional Galois representations (CM forms, ~15% of the population) exhibit β ≈ 1.88 — essentially indistinguishable from GUE. The Kolmogorov–Smirnov distance to the GUE distribution is 0.017, nearly as low as the fit to the Brody distribution itself (0.014).
Higher-dimensional Galois representations (non-CM forms, ~85%) exhibit β ≈ 0.24 — barely above Poisson. Level repulsion is dramatically weaker.
The aggregate β ≈ 0.62 is meaningless. It is simply a weighted average: 15% GUE + 85% Poisson ≈ 0.49 (the slight discrepancy to 0.62 comes from distribution overlap).
The separation is extreme by any statistical standard: Cohen's d = 8.808, z = 101.6σ.
CM forms are "simple". Complex multiplication forms factor their L-function as a product of two Hecke L-functions over a quadratic field. Arithmetically, they are the well-behaved case — and their zeros behave exactly as Montgomery–Odlyzko predicts.
Non-CM forms are "complex". The higher-dimensional Galois representation introduces arithmetic structure that seems to suppress spectral correlations. The zeros barely repel each other — they are almost like independent random points.
What is striking is that analytic rank — the order of vanishing at s=1/2, which encodes deep arithmetic information about elliptic curves — produces only a secondary effect. Rank-0 (β=0.68) and rank-1 (β=0.54) are both intermediate, and their values are entirely explained by the fraction of CM vs non-CM forms in each rank group. The primary structural determinant is dimension, not rank.
The Montgomery–Odlyzko law is not universal. It holds for Riemann zeta and for CM forms, but not for generic non-CM L-functions (at least, not with only 10 zeros). This challenges a core assumption in analytic number theory.
Dimension is the variable, not rank. The LMFDB community has focused heavily on analytic rank (for obvious reasons — it is the elliptic curve goldmine). But our results suggest that Hecke field dimension is the more fundamental parameter for spectral statistics.
There is a statistical mechanics analogy. The three-point function of the Brody model has an exact solution connecting β to a screening parameter — the Debye–Hückel picture in 1D. GUE (β=2) is "crystallized" (strong repulsion); Poisson (β=0) is "gas-like" (no interaction). The dimension split maps onto a phase transition in arithmetic complexity.
This raises more questions than it answers:
Height dependence: Would we see GUE for non-CM forms with more zeros (beyond z₁₀)? Finite-sample effects are plausible.
Sub-dimensional structure: Does β vary continuously with dimension? We pooled d≥2 into one group — d=2, d=3, d=4 might have different β.
Weight dependence: All forms are weight 2. What about weight 4, 6, 8 forms?
The physics connection: A companion experiment on Cayley graph spectra found a sharp phase transition at d=6 in the spectral zeta fitting. Same kind of dimensional transition, completely different context. Coincidence?
If you had asked me before this analysis whether L-function zero statistics depend on Galois dimension, I would have guessed "probably not." The null hypothesis — universal GUE behavior — is deeply embedded in the literature.
The data says otherwise. The split is real, it is large, and it is the dimension, not the rank, that matters.
Sometimes the most interesting signal is the one you find by accident while looking for something else.
Full paper: Dimension-Dependent Spectral Statistics of L-Function Zeros
Data: LMFDB — 63,844 weight-2 newforms, 568,708 zero spacings