Murmurations are a recently discovered statistical pattern in the arithmetic of elliptic curves. When you plot the average Frobenius traces of elliptic curves against the prime, grouped by analytic rank, the averages don’t converge monotonically — they oscillate, creating wave-like patterns that are visible in the data but not yet fully explained theoretically.

Convolutional neural networks, trained on Frobenius trace sequences, can predict the analytic rank of an elliptic curve with high accuracy. The network sees only the local arithmetic data — the traces at small primes — and from this deduces a global property of the curve. This is not surprising in principle (the Birch–Swinnerton-Dyer conjecture asserts that local data determines global behavior), but the networks learn the relationship without being told the conjecture.

The interpretive tool is saliency analysis: which inputs matter most for the network’s predictions. The saliency maps reveal that the network attends to exactly the primes and trace patterns that exhibit murmuration behavior. The neural network independently rediscovered the statistical structure that mathematicians found by visual inspection of data, confirming that murmurations carry genuine predictive information about rank.

The connection between murmurations and machine learning is mutually illuminating. Murmurations, discovered in 2022, showed that arithmetic statistics contains more structure than previously appreciated. The neural network analysis shows that this structure is not merely aesthetic but functionally predictive — the oscillations encode rank.

The network doesn’t know the Birch–Swinnerton-Dyer conjecture. It just learned to read the data, and the data oscillates in a pattern that encodes exactly what the conjecture predicts it should.