Predicting L-Function Properties from Trace-Index Graphs using Graph Neural Networks
AI & Machine Learning
graph-neural-networksnumber-theorylmfdbspectral-filtersmodular-formssato-tatemachine-learningriemann-hypothesis
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This paper is the positive counterpart to our negative-result study: When Graph Neural Networks Meet the Riemann Hypothesis: A Systematic Negative Study
Can machine learning predict arithmetic properties of modular forms by operating on graph-structured representations of Fourier coefficient data? We investigate this question using Graph Neural Networks on trace-index graphs: 1000-node graph representations of individual newforms, where each node corresponds to a Fourier index and edges encode sequential adjacency, primality structure, and k-nearest-neighbor similarity in coefficient space.
On 46,347 weight-2 newforms from the LMFDB, a 3-layer Chebyshev spectral filter network (K=5) predicts:
Spectral filters consistently outperform plain GCN, with the largest gains on rare-class detection (+38.87 pp in class-2 F₁). Cross-level generalization shows regression degrades modestly (-14% in R²) while classification suffers more severely. Our Sato-Tate moment analysis across 53,779 forms confirms the empirical trace distribution matches SU(2) theory, with CM forms clearly distinguished by their U(1) moments.
| Target | Metric | GCN Baseline | ChebConv K=5 |
|---|---|---|---|
| z₁ (first L-function zero) | R² | 0.559 | 0.625 |
| Analytic rank (3-class) | Accuracy | 91.27% | 94.16% |
| Analytic rank | F₁ macro | 74.61% | 89.22% |
| Analytic rank (class ≥2) | F₁ | 40.00% | 78.87% |
| CM status (binary) | Accuracy | 99.96% | 100.00% |
This work follows a systematic negative result: GNNs on Cayley graphs of SL(2,Fₚ) cannot predict spectral properties because vertex-transitivity forces every node to be structurally identical. A message-passing GNN receives the same information from every node, making graph-level prediction impossible. Across seven experiment tracks, every GNN configuration failed or provided only marginal improvement over baselines.
The solution: instead of constructing graphs from algebraic structure (the group SL(2,Fₚ)), construct them from Fourier coefficient data itself. This gives the GNN:
The paper includes a statistical analysis of normalized Hecke traces xₚ = aₚ(f) / (2√p) across 53,779 forms:
This connects two central themes in the analytic theory of modular forms: the equidistribution of Hecke eigenvalues and L-function zero statistics.
For each modular form f, we construct a graph G_f with:
All models trained with AdamW, CosineAnnealingLR, early stopping, stratified 80/10/10 split.
Training on conductors ≤ 3000 (low level) and testing on conductors > 4000 (high level) reveals an asymmetry:
This suggests the GNN learns conductor-independent patterns for zero prediction but conductor-dependent patterns for rare-class rank classification.
A follow-up study of 63,844 weight-2 newforms from the LMFDB reveals a sharp, dimension-dependent transition in L-function zero-spacing statistics — empirical evidence for the Katz–Sarnak philosophy that families of L-functions carry random-matrix-type symmetry.
Key findings:
| Form dimension | Spacing ensemble | Brody β |
|---|---|---|
| dim = 1 | GUE | 1.88 |
| dim ≥ 2 | near-Poisson | 0.24 |
| dim ≥ 2 (outliers, ~6%) | GUE (retained) | — |
This complements the trace-index GNN work: where the GNN learns to predict individual zero locations and arithmetic properties from coefficient structure, this study characterizes the global spacing law and shows it is governed almost entirely by scalar invariants (dimension, level) rather than the detailed Hecke data. The convergence on "scalar metadata beats raw traces" is a recurring theme across the whole research program.
Published: The Two-Population Structure of L-Function Zero Spacings: dim=1 to GUE, dim≥2 to Poisson with 6% Outliers (Zenodo 10.5281/zenodo.21979592)
Building on the trace-index graph work, a new study achieves F1=0.900 for CM detection in weight-2 newforms using Gradient Boosting Machines trained Prime-indexed Fourier coefficients combined with 11 Sato-Tate moments:
Key insight: The M₄/M₂ dimensional dilation ratio captures discriminative information about CM structure, while individual trace coefficients at p=23, 41, and 7 provide complementary signal. This demonstrates that CM is learnable from small-dimensional feature sets without expensive Elliptic Curve analysis.
Published: Data-Driven Detection of Complex Multiplication in Weight 2 Cusp Forms (Zenodo 10.5281/zenodo.20555502)
This work follows a systematic negative result: GNNs on Cayley graphs of SL(2,Fₚ) cannot predict spectral properties because vertex-transitivity forces every node to be structurally identical. A message-passing GNN receives the same information from every node, making graph-level prediction impossible. Across seven experiment tracks, every GNN configuration failed or provided only marginal improvement over baselines.
The solution: instead of constructing graphs from algebraic structure (the group SL(2,Fₚ)), construct them from Fourier coefficient data itself. This gives the GNN: