$L$-FUNCTIONS OF ELLIPTIC CURVES AND BINARY RECURRENCES. Issue 3 (22nd March 2013) - LUCA, FLORIAN; OYONO, ROGER; YALCINER, AYNUR
Abstract: Let $L(s, E)= {\mathop{\sum }\nolimits}_{n≥ 1} {a}_{n} {n}^{- s} $ be the $L$ -series corresponding to an elliptic curve $E$ defined over $ \mathbb{Q} $ and $u = \mathop{\{ {u}_{m} \} }\nolimits_{m≥ 0} $ be a nondegenerate binary recurrence sequence. We prove that if ${ \mathcal{M} }_{E} $ is the set of $n$ such that ${a}_{n} \not = 0$ and ${ \mathcal{N} }_{E} $ is the subset of $n\in { \mathcal{M} }_{E} $ such that $\vert {a}_{n} \vert = \vert {u}_{m} \vert $ holds with some integer $m≥ 0$, then ${ \mathcal{N} }_{E} $ is of density $0$ as a subset of ${ \mathcal{M} }_{E} $ .