Elliptic curves and spin

Abstract

In the early 2000s, Ramakrishna asked the question: for the elliptic curve E : y2 =x3 -x, what is the density of primes p for which the Fourier coefficient ap(E) is a cube modulo p? As a generalisation of this question,Weston Zaurova formulated conjectures concerning the distribution of power residues of degree m of the Fourier coefficients of elliptic curves E/Q with complex multiplication. In this paper, we prove the conjecture of Weston Zaurova for cubic residues using the analytic theory of spin. Our proof works for all elliptic curves E with complex multiplication.