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$L$-FUNCTIONS OF ELLIPTIC CURVES AND BINARY RECURRENCES

Part of: Arithmetic algebraic geometry Sequences and sets Multiplicative number theory

Published online by Cambridge University Press:  22 March 2013

FLORIAN LUCA ,
ROGER OYONO  and
AYNUR YALCINER
FLORIAN LUCA*
Affiliation:
Fundación Marcos Moshinsky, Instituto de Ciencias Nucleares UNAM, Circuito Exterior, C.U., Apdo. Postal 70-543, Mexico D.F. 04510, Mexico
ROGER OYONO
Affiliation:
Équipe GAATI, Université de la Polynésie Française, BP 6570, 98702 Faa’a, Tahiti, French Polynesia email [email protected]
AYNUR YALCINER
Affiliation:
Department of Mathematics, Faculty of Science, Selçuk University, Campus 42075 Konya, Turkey email [email protected]
*
✉[email protected]
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Abstract

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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} $.

Keywords

L-functions of elliptic curveslinear recurrence sequences

MSC classification

Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11N36: Applications of sieve methods
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 509 - 519
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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