\(L\)-functions of elliptic curves and binary recurrences (Q2872021)

Let \(E\) be an elliptic curve defined over the rational number field, with Hasse-Weil \(L\)-function \(L(E,s) = \sum_{n \geq 1} a_n n^{-1}\). Let \(\mathbf{u} = \{ u_{m} \}_{m \geq 0}\) be a non degenerate binary recurrence sequence satisfying the relation NEWLINE\[NEWLINEu_{m+2} = ru_{m+1} + su_m \text{ for all \(m \geq 0\),}NEWLINE\]NEWLINE where \(r\) and \(s\) are integers for which the quadratic polynomial \(x^2 -rx +s = 0\) has distinct roots \(\alpha\) and \(\beta\). The authors estimate the density of the subset of all nonvanishing coefficients \(a_n\) of \(L(E,s)\) which coincide with some \(u_m \in {\mathbf{u}}\). To be more precise, let NEWLINE\[NEWLINEM_E = \{ n \in{\mathbb{Z}}_{\geq 1} : a_n \neq 0 \},NEWLINE\]NEWLINE and let NEWLINE\[NEWLINEN_E = \{ n \in M_E: | a_n | = | u_m | \text{~for some \(m \geq 0\)} \} . NEWLINE\]NEWLINE Given a positive real number \(x\) and a subset \(A\) of the positive integers, let us write \(A(x) = A \cap [1, x]\). The authors determine that there exists a positive constant \(c = c(E, {\mathbf{u}}) \geq 0\) depending on \(E\) and \(\mathbf{u}\) such that for all \(x \geq 2\), NEWLINE\[NEWLINE| N_E(x) | = O_E \left( \frac{| M_E(x)|}{ (\log x)^c }\right).NEWLINE\]NEWLINE The authors also provide in the final section some heuristics for their conjecture that there exists a constant \(\eta > 0\) for which \(| N_E(x) | = O_E(x^{1 - \eta})\).