Machine learning the vanishing order of rational L-functions

The Big Picture

Imagine trying to predict the size of a hidden treasure chest by studying only the ripples it makes on the surface of a vast mathematical ocean. That’s roughly the challenge of determining the rank of an elliptic curve, one of the deepest unsolved problems in number theory.

An elliptic curve isn’t the oval you might picture. It’s a family of equations of the form y² = x³ + ax + b whose solutions form a curve with rich algebraic structure. The rank counts how many fundamentally independent solutions exist using only fractions (what mathematicians call rational points). Computing it directly is notoriously difficult.

So mathematicians study a proxy: an L-function, an infinite formula built from the curve’s coefficients that encodes its arithmetic properties. Think of it as a fingerprint, a structured sequence of numbers that carries hidden information about the original curve.

The link between rank and L-functions is the Birch and Swinnerton-Dyer (BSD) conjecture, one of the Millennium Prize Problems. BSD predicts that the rank equals the vanishing order of the L-function at a special input called the central point: how many times the function “touches zero” there. Vanishing order zero means finitely many rational points; higher orders mean more. Computing this vanishing order, though, requires delicate analytic techniques.

What if a machine could learn to read it directly from the raw coefficients?

A team of nine researchers set out to test that idea across a collection of 248,359 rational L-functions, and found that machine learning methods predict vanishing order with high accuracy.

Key Insight: Neural networks and classical ML methods can accurately predict the vanishing order of rational L-functions from their Dirichlet coefficients alone, with direct implications for BSD.

How It Works

The researchers built a dataset called RAT, drawn from the L-functions and Modular Forms Database (LMFDB). It spans five families of mathematical objects:

• ECQ: elliptic curves over Q
• ECNF: elliptic curves over number fields
• CMF: classical modular forms
• DIR: Dirichlet characters
• G2Q: genus 2 curves

In total, the dataset contains 248,359 L-functions with root analytic conductor below 4, a complexity measure that keeps the dataset balanced across origins.

Each L-function arrives as a sequence of Dirichlet coefficients ${a_n}$, numbers associated with each prime that capture how the underlying mathematical object behaves locally. The team fed these sequences directly into their models as feature vectors, working in the arithmetic normalization where coefficients are whole-number-like algebraic quantities and the functional equation takes a clean symmetric form.

Before training any neural network, the team probed for structure with simpler tools.

Principal Component Analysis (PCA) compresses high-dimensional data down to its most informative directions. Projecting coefficient vectors onto the top two principal components revealed something striking: L-functions naturally clustered by vanishing order, no labels required. The geometry of coefficient space itself separates rank-0 from rank-1 from rank-2 curves.

Linear Discriminant Analysis (LDA), a supervised method that maximizes separation between labeled classes, achieved strong classification accuracy, confirming that the PCA clusters carry genuine predictive signal. Feed-forward and convolutional neural networks pushed accuracy further, learning nonlinear features from coefficient sequences that LDA missed.

Then there are the murmurations. Named after the coordinated flight of starling flocks, this phenomenon was first observed in 2022 in elliptic curves over Q. When you average the Dirichlet coefficients of many L-functions at the same prime, organized by root analytic conductor, the averages don’t wash out to zero. Instead they trace coherent oscillating patterns that differ systematically between vanishing orders.

The team found murmuration-like signatures across all five sub-datasets, which suggests the phenomenon extends well beyond its original setting.

The researchers also tested transfer learning: training a model on one sub-dataset and applying it to another. This probes whether the learned representations capture something universal about L-functions or something specific to a single mathematical family. Transfer between families worked to a meaningful degree, hinting that the networks pick up on genuine analytic structure shared across rational L-functions more broadly.

Why It Matters

The BSD conjecture and its generalizations (the Beilinson-Bloch-Kato conjectures) are central to modern number theory. They tie the analytic properties of L-functions to deep algebraic invariants. Proving BSD remains one of the hardest open problems in mathematics.

What this paper shows is that the information needed to determine vanishing order is already present in the first few hundred Dirichlet coefficients, in a form machines can read. That’s not a proof of BSD. But it is strong empirical evidence that the conjecture is correct, and that the structure it predicts is detectable by relatively simple methods.

The murmuration phenomenon was itself first spotted by researchers who noticed a pattern in averages before anyone had a theoretical explanation. By scaling up to a quarter million L-functions spanning multiple degrees and origins, this team has shown murmurations and related structure persist across mathematical families. The patterns these networks learn may point toward new conjectures about which arithmetic invariants control vanishing order.

Bottom Line: Machine learning doesn’t just classify L-functions; it reveals that the rank of an arithmetic object is already encoded as detectable geometric structure in the first few hundred terms of its associated L-function. This is empirical evidence for one of number theory’s deepest conjectures, read off by a neural network.