Y. He

K. Lee

T. Oliver

Y. Qi

Computer vision distinguishes elliptic curves from random data and predicts analytic rank by reading arithmetic patterns in digital images.

Inspired by the Langlands programme, elliptic curves and their twists are encoded as digital images. A two-dimensional convolutional neural network distinguishes their arithmetic structure from Sato–Tate random data far better than one-dimensional trace data. The same vision architecture can predict an elliptic curve’s analytic rank, showing how ideas from computer vision can expose hidden number-theoretic structure. 

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.

Arithmetic as images

Computer vision and converse theorems