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.
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: