Sequences generated by elliptic curves

Betül Gezer ¹   ; Osman Bizim ¹

¹ Department of Mathematics Faculty of Science Bursa Uludağ University Görükle, 16059, Bursa, Turkey

Acta Arithmetica, Tome 188 (2019) no. 3, pp. 253-268

Résumé

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.

Détail

DOI : 10.4064/aa170504-25-6

Keywords: study properties sequences geq geq generated numerators x y coordinates multiples point elliptic curve defined field prove defined finite field these sequences purely periodic generalize result modulo prime powers consequence deduce certain subsequences these sequences converge p adically mathbb cauchy