Numerical evidence on the uniform distribution of power residues for elliptic curves (Q1036242)

This article offers a light introduction to the theory of elliptic curves: on the first few pages, Bézout's theorem, the group law, the Hasse-Weil bound and curves with complex multiplication are introduced. The second part describes numerical results supporting the following conjecture made by Ramakrishna: let \(E\) be an elliptic curve defined over \(\mathbb Q\). Pick an even number \(M\), let \(r_1, \dots, r_s\) denote the coprime residue classes modulo \(M\), and let \(R_i\) denote the set of primes \(p \equiv r_i \bmod M\) less than some bound \(n\) for which the number \(N_p\) of \({\mathbb F}_p\)-rational points on \(E\) is a quadratic residue modulo \(p\). If \(E\) does not have CM, it is believed that \(\lim_{n \to \infty} \# R_i / \# R_j = 1\).