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

TL;DR

This paper investigates the existence of arbitrarily long sequences of primes for which the number of points on an elliptic curve modulo these primes satisfy a specific multiplicative relation, under certain hypotheses.

Contribution

It proves that for any fixed number of primes greater than or equal to three, there are infinitely many such prime tuples assuming the Generalized Riemann Hypothesis, and conjectures this extends to shorter sequences.

Findings

Proves infinite occurrence of prime tuples for k≥3 under GRH.

Conjectures similar results for k=1,2.

Supports the existence of elliptic progressions of primes.

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 $p_1<p_2<\ldots <p_k$ 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 $\varlimsup_{n\to\infty} G_k(E,n)=\infty$ for any integer $k\geq 3$. 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 $N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)$.