Predicting L-Function Properties from Trace-Index Graphs using Graph Neural Networks

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Abstract

We investigate whether graph neural networks can predict arithmetic properties of modular forms by operating on graphs constructed from Fourier coefficient data. For each modular form $f$, we build a trace-index graph: 1000 nodes (one per index $n = 1, \ldots, 1000$, corresponding to Fourier coefficients $a_n(f)$), connected by sequential, primality, and $k$-nearest-neighbor edges, with node features encoding the coefficient values and their normalizations. On 46,347 weight-2 newforms from the LMFDB, a 3-layer Chebyshev convolution network ($K=5$) predicts the first $L$-function zero with $R^2 = 0.625$, analytic rank with 94.16% accuracy, and CM status with 100% accuracy.

Key Results

Target	Metric	GCN Baseline	ChebConv $K{=}5$
$z_1$ (first L-function zero)	$R^2$	0.559	0.625
Analytic rank (3-class)	Accuracy	91.27%	94.16%
Analytic rank	$F_1$ macro	74.61%	89.22%
Analytic rank (class ≥ 2)	$F_1$	40.00%	78.87%
CM status (binary)	Accuracy	99.96%	100.00%

Approach

Trace-Index Graph Construction

For each modular form $f$, we construct a graph $G_f$ with:

• 1000 nodes — one per index $n = 1, \ldots, 1000$, each carrying 5-dimensional features: $(\log(n),\; a_n(f),\; a_n(f)/(2\sqrt{n}),\; \mathbf{1}[n \text{ is prime}],\; \sqrt{n})$
• ~9,500 edges from three sources:
  1. Sequential: $(i, i+1)$ for consecutive indices
  2. Prime: $(i, j)$ when both are prime (168 prime-indexed nodes)
  3. $k$NN: $(i, j)$ when $j$ is among the $k{=}3$ nearest indices in coefficient-value space

This is fundamentally different from prior work using Cayley graphs of $\text{SL}(2, \mathbb{F}_p)$, which are vertex-transitive and give GNNs no local diversity to exploit.

Why This Works (and Cayley Graphs Don't)

Property	Cayley $\text{SL}(2, \mathbb{F}_p)$	Trace-Index
Vertex-transitive	Yes	No
Node features	Identical (structural)	Unique (Fourier coefficients)
Graph topology	Algebraic (group)	Data-driven ($k$NN)
Best $R^2$ (test)	$<0$ (all experiments)	0.625

Cross-Level Generalization

Training on conductors ≤ 3000 and testing on conductors > 4000 reveals an interesting asymmetry:

• Regression generalizes well: $z_1$ $R^2$ drops only 14% (0.625 → 0.538)
• Classification degrades: Rank accuracy drops from 94.16% to 87.58%, and rare class 2 $F_1$ collapses from 78.87% to 25.66%

This suggests the GNN learns conductor-independent patterns for regression but conductor-dependent patterns for classification.

Limitations

1. Below-sklearn regression: $R^2 = 0.625$ is lower than tree ensembles on raw Fourier coefficients ($R^2$ 0.73–0.96)
2. Weight-2 only: Generalization to other weights is untested
3. Rare-class sensitivity: Class 2 $F_1$ collapses on unseen conductors
4. No causal claims: The GNN learns statistical patterns, not proofs of arithmetic theorems

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Data source: The L-Functions and Modular Forms Database (LMFDB)