Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curves

The Big Picture

Imagine trying to predict whether a tangled knot can be untied using only a blurry photograph taken from a strange angle. That’s roughly the challenge mathematicians face when computing the rank of an elliptic curve.

Despite the name, an elliptic curve has nothing to do with ellipses. It’s a special type of equation, something like y² = x³ + ax + b, whose solutions trace out a smooth curve with rich algebraic structure. The rank is a single whole number counting how many fundamentally independent fraction-valued solutions the curve has. It’s notoriously hard to compute, yet it encodes deep truths about the arithmetic of these equations.

A neural network, the same kind that recognizes faces and translates languages, can predict that rank with high accuracy just by scanning a list of numbers called Frobenius traces. These traces are relatively simple to compute: for each prime p, count how many solutions the curve has in a stripped-down number system with finitely many values, then subtract p + 1. The result, a_p, is a small integer.

Feed a thousand of these into a neural network and it somehow knows the rank. But why does it work? A team of eight mathematicians and physicists set out to answer that question, and they found two distinct arithmetic phenomena hiding inside those innocent-looking traces.

Key Insight: When a neural network learns to predict elliptic curve rank, it implicitly discovers two distinct arithmetic structures, “murmurations” and Mestre–Nagao sums, and the balance between them shifts as training progresses and varies with the arithmetic complexity of the curve.

How It Works

The researchers built a dataset called XECQ: over 1.5 million elliptic curves over the rational numbers, each with conductor (a measure of arithmetic complexity) no greater than 400,000 and rank between 0 and 4. Each curve was represented as 1,229 Frobenius traces a_p for all primes up to 10,000, normalized to the interval [−1, 1].

They trained a one-dimensional convolutional neural network (1D CNN), an architecture that detects local patterns across a sequence of numbers, the same technique used in audio processing and time-series analysis. The network is compact: three convolutional layers (16, 32, and 64 channels, kernel size 3), followed by two fully connected layers of 128 neurons. Its job is to look at those 1,229 numbers and output a predicted rank. It does so with high accuracy, matching results from earlier experiments by Kazalicki–Vlah and Pozdnyakov.

The real payoff comes in the interpretation. The team used saliency curves, a tool from explainability research that measures how much each input value influences the network’s output. If you nudge a_p slightly, how much does the predicted rank change? Plotting these sensitivities across all primes reveals which parts of the input the network actually attends to.

Two patterns showed up, and they appeared at different stages of training:

• Early in training, saliency curves show clear murmuration patterns: oscillating, wavelike signatures that sweep across the prime axis like a flock of starlings. These murmurations, first discovered in 2022, are a genuine number-theoretic phenomenon where Frobenius traces at certain primes carry systematic rank-dependent signals.
• Late in training, the oscillations give way to a smooth decay closely matching Mestre–Nagao sums, a classical tool where each prime p is weighted roughly by log(p)/p, and their weighted sum converges to a value tied to the rank.

The network begins by learning to “listen” to the rhythm of murmurations, then gradually shifts to the slower, more monotonic signal encoded in Mestre–Nagao sums.

Why It Matters

Not all elliptic curves are created equal, and the conductor tells you how arithmetically complicated a curve is. The researchers split their data into four conductor windows (roughly 0–10,000, 100,000–110,000, 200,000–210,000, and 300,000–310,000) and trained separate networks on each.

The result is one of the paper’s most concrete findings: murmurations matter much more for curves with small conductor. For low-conductor curves, saliency clearly exhibits murmuration-like oscillations. For high-conductor curves, those oscillations are suppressed and Mestre–Nagao-type decay dominates almost immediately. The network adapts its strategy depending on what arithmetic structure is most accessible in the data.

There’s an additional twist. Even among small-conductor curves, where murmurations are most prominent, the murmuration patterns appear only for rank 1 curves. For rank > 1, they vanish even in early training. Murmurations are not a universal feature of rank prediction; they’re a signature specifically tied to how rank-1 curves differ from rank-0 curves.

This work is simultaneously a machine learning interpretability study and a piece of number theory. The researchers aren’t just showing that neural networks can predict rank. They’re using the network as a lens to understand which primes carry the most information and why certain arithmetic invariants are detectable at all.

The broader implication reaches toward the Langlands program, the sweeping mathematical vision connecting number theory, geometry, and analysis through deep relationships between L-functions, Galois representations, and automorphic forms. Murmurations appear to be a concrete, experimentally accessible fingerprint of these abstract connections. Saliency analysis could serve as a kind of telescope pointed at structural patterns that pure mathematics hasn’t yet fully explained.

Open questions remain. Why do murmurations vanish for rank > 1? Can saliency-based interpretability reveal similar structures for other L-functions or for abelian varieties (higher-dimensional generalizations of elliptic curves)? Can the patterns uncovered here guide rigorous proofs about Frobenius trace distributions?

Bottom Line: A neural network trained to predict elliptic curve rank doesn’t just memorize patterns. It rediscovers two known arithmetic phenomena in sequence, with the balance between them controlled by conductor, rank, and training epoch. Machine learning is becoming a telescope for number theory.

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