On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$

Abstract

For a given elliptic curve $E/\mathbb{Q}$, let $N_p(E)$ be the number of points on $E$ modulo $p$ for a prime of good reduction for $E$. Given integer $n$, let $G_k(E,n)$ be the number of $k$-tuples of $p_1<p_2<\dots <p_k$ primes of good reduction for $E$, for which the equation in the title holds, then on assuming the Generalized Riemann Hypothesis for elliptic curves without CM (and unconditionally if the curves have complex multiplication), I show that ${\overline{\lim }}_{n\to \infty }{G}_k(E,n)=\infty$ for any integer $k\ge 3$. I conjecture that this result also holds for $k=1,2$ i.e. this conjecture says that there are arbitrarily long ``elliptic progressions of primes'' i.e. sequences of primes $p_1<p_2<\cdots <p_m$ of arbitrary lengths $m$ such that $N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)$.

35 Pages. Substantially expanded and completely revised version. To appear in The Ramanujan Journal