$L$-FUNCTIONS OF ELLIPTIC CURVES AND BINARY RECURRENCES. Issue 3 (22nd March 2013)

• Record Type: Journal Article
• Authors: LUCA, FLORIAN; OYONO, ROGER; YALCINER, AYNUR
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 88:Issue 3(2013)
• Page Start: 509
• Page End: 519
• Publication Date: 2013-03-22
• Subjects: primary 11G40, -- secondary 11B39, -- 11N36; L-functions of elliptic curves, -- linear recurrence sequences; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972713000166
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract: Let $L(s, E)= {\mathop{\sum }\nolimits}_{n\geq 1} {a}_{n} {n}^{- s} $ be the $L$ -series corresponding to an elliptic curve $E$ defined over $ \mathbb{Q} $ and $\mathbf{u} = \mathop{\{ {u}_{m} \} }\nolimits_{m\geq 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 $m\geq 0$, then ${ \mathcal{N} }_{E} $ is of density $0$ as a subset of ${ \mathcal{M} }_{E} $ .