Squares in a certain sequence related to \(L\)-functions of elliptic curves (Q1946678)

Let \(E\) be an elliptic curve defined over the rationals, and write its \(L\)-function by \(L(s,E):=\sum_{n\geq1}a_n n^{-s}\) for \(\text{Re}(s)>3/2\). Recall for \(p\) primes not dividing the discriminant \(\Delta\) of the elliptic curve \(E\), we have that \(p+1-a_p\) is the number of elements of the set of \(\mathbb{F}_p\)-points of the reduction at \(p\) of the elliptic curve \(E\), and for the primes \(p\) dividing \(\Delta\) then \(a_p\in\{-1,0,1\}\). For \(\text{Re}(s)>3/2\) we have the equality \[ L(s,E)=\prod_{p|\Delta}\frac{1}{1-a_p p^{-s}}\prod_{p\nmid \Delta}\frac{1}{1-a_p p^{-s}+p^{1-2s}}. \] Consider the set \[ \mathcal{N}_E:=\{n: n^2-a_{n^2}+1\text{ is\;a\;square}\}. \] Under the assumption that \(E\) has no complex multiplication and that it satisfies Sato-Tate conjecture, the authors of the paper under review obtain the estimate \[ \#(\mathcal{N}_E\cap [1,x])=:\mathcal{N}_E(x)=O(\frac{x}{(\log x)^{0,00001}}) \] for all \(x\geq 2\), obtaining in particular that \(\mathcal{N}_E(x)\) has asymptotic density 0.