## Sequences generated by elliptic curves

### Volume 188 / 2019

Acta Arithmetica 188 (2019), 253-268
MSC: Primary 14H52; Secondary 11G07, 14G20, 11B37.
DOI: 10.4064/aa170504-25-6
Published online: 7 March 2019

#### Abstract

We study the properties of the sequences $(G_{n}(P))_{n\geq 0} $ and $(H_{n}(P))_{n\geq 0}$ generated by the numerators of the $x$- and $y$-coordinates of the multiples of a point $P$ on an elliptic curve $% E$ defined over a field $K$. We prove that if $E$ is defined over a finite field, then these sequences are purely periodic. Then we generalize this result to the case of modulo prime powers. As a consequence, we deduce that certain subsequences of these sequences converge $p$-adically, i.e., are $\mathbb{Z}_{p}$-Cauchy.