Computer vision and converse theorems

TL;DR

This paper explores how convolutional neural networks can distinguish between arithmetic data from elliptic curves and random matrix data, inspired by converse theorems in the Langlands program.

Contribution

It introduces a novel approach of encoding elliptic curves as images for CNN classification and demonstrates improved separation of data types and rank prediction.

Findings

2D CNN outperforms 1D CNN in data separation

CNN can predict elliptic curve analytic rank

Encoding elliptic curves as images enhances classification

Abstract

Random matrices provide a well-established statistical model for a range of arithmetic phenomena. In this paper, we investigate the extent to which one- and two-dimensional convolutional neural networks (CNNs) can distinguish between arithmetic data arising from elliptic curves with conductor in a fixed interval and random matrix data drawn from the same Sato-Tate distribution. Inspired by converse theorems in the Langlands program, we represent each elliptic curve together with its twists as a vector field and, subsequently, encode that vector field as a digital image. We observe that a two-dimensional CNN trained on this image data is better able to separate conductor families from random matrix data than a one-dimensional CNN trained on vectors of Frobenius traces without twisting data. We also observe that the same two-dimensional architecture can predict the analytic rank of an elliptic curve, and it does so by factoring through the (untwisted) Frobenius traces.