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$L$ -FUNCTIONS OF ELLIPTIC CURVES AND BINARY RECURRENCES

  • FLORIAN LUCA (a1), ROGER OYONO (a2) and AYNUR YALCINER (a3)

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} $ .

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References

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$L$ -FUNCTIONS OF ELLIPTIC CURVES AND BINARY RECURRENCES

  • FLORIAN LUCA (a1), ROGER OYONO (a2) and AYNUR YALCINER (a3)

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