Elliptic curves and spin

TL;DR

This paper proves conjectures about the distribution of cubic power residues of Fourier coefficients of elliptic curves with complex multiplication, using the analytic theory of spin, thus advancing understanding of prime distributions related to elliptic curves.

Contribution

It provides a proof of Weston--Zaurova's conjecture on cubic residues of Fourier coefficients for all CM elliptic curves using the analytic theory of spin.

Findings

Proves conjecture for cubic residues of Fourier coefficients.

Applies analytic theory of spin to elliptic curves with CM.

Results hold for all elliptic curves with complex multiplication.

Abstract

In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As a generalization of this question, Weston--Zaurova formulated conjectures concerning the distribution of power residues of degree $m$ of the Fourier coefficients of elliptic curves $E/\mathbb{Q}$ with complex multiplication. In this paper, we prove their conjecture for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves $E$ with complex multiplication.