Murmurations: a case study in AI-assisted mathematics

TL;DR

An overview of murmurations is presented, contextualizing them within number theory and AI, and finding that they connect to central themes surrounding the conjecture of Birch and Swinnerton-Dyer and perspectives from random matrix theory.

Abstract

We report the emergence of a striking new phenomenon in arithmetic, which we call murmurations. First observed experimentally through averages over large arithmetic datasets, murmurations can be detected and analyzed using standard interpretability tools from machine learning, including principal component weightings, saliency curves, and convolutional filters. Although discovered computationally, they constitute a genuinely new and intriguing phenomenon in arithmetic that can be formulated and investigated using established tools of number theory. In particular, murmurations encode subtle information about Frobenius traces and naturally belong to the framework of arithmetic statistics. More precisely, murmurations connect to central themes surrounding the conjecture of Birch and Swinnerton-Dyer and perspectives from random matrix theory. In this paper, we present an overview of murmurations, contextualizing them within number theory and AI.

Figures (7)

• Figure 1: A plot of the primes ($x$-axis) against the aggregate discrepancy ($y$-axis).
• Figure 2: Graphs for conic sections defined by the quadratic equations (left) $3x^2+5y^2=12$ and (right) $3x^2-5y^2=12$.
• Figure 3: Graphs for elliptic curves defined by the cubic equations (left) $y^2+y=x^3+x^2+x$ and (right) $y^2+y=x^3-x$.
• Figure 4: (Top) A plot showing how the discrepancy varies, with $y^2+y=x^3-x$ in red and $y^2+y=x^3+x^2+x$ in blue. (Bottom) A plot showing how the aggregate discrepancy varies for the same curves.
• Figure 5: (Left) The curve $y^2=x^3$ has a cuspidal singularity, and (right) the curve $y^2=x^3+x^2$ has a nodal singularity.