Distribution of squarefree values of sequences associated with elliptic curves

TL;DR

This paper investigates the frequency of squarefree values in sequences derived from elliptic curves over finite fields, providing bounds and conjectural asymptotics for their distribution across primes.

Contribution

It offers new bounds for the occurrence of squarefree values in sequences associated with elliptic curves and supports a conjectural asymptotic distribution over families of curves.

Findings

Upper bounds for primes p with squarefree sequence values

Consistency of conjectural asymptotics with average over curves

Analysis of specific sequences related to elliptic curve reductions

Abstract

Let E be a non-CM elliptic curve defined over Q. For each prime p of good reduction, E reduces to a curve E_p over the finite field F_p. For a given squarefree polynomial f(x,y), we examine the sequences f_p(E) := f(a_p(E), p), whose values are associated with the reduction of E over F_p. We are particularly interested in two sequences: f_p(E) =p + 1 - a_p(E) and f_p(E) = a_p(E)^2 - 4p. We present two results towards the goal of determining how often the values in a given sequence are squarefree. First, for any fixed curve E, we give an upper bound for the number of primes p up to X for which f_p(E) is squarefree. Moreover, we show that the conjectural asymptotic for the prime counting function \pi_{E,f}^{SF}(X) := #{p \leq X: f_p(E) is squarefree} is consistent with the asymptotic for the average over curves E in a suitable box.