L-Functions of elliptic curves and binary recurrences

Let L(s; E) = Σn≤1ann-s be the L-series corresponding to an elliptic curve E defined over ℚ and u = {um}m≥0 be a nondegenerate binary recurrence sequence. We prove that if ME is the set of n such that an , 0 and NE is the subset of n ε ME such that |an| = |um| holds with some integer m ≤ 0, then NE is of density 0 as a subset of ME. © 2012 Australian Mathematical Publishing Association Inc.