arXiv subjects

Betül Gezer

Publications and source records attributed to Betül Gezer.

3 recordsLinked to original sources

arXiv2026

p-adic Properties of Translated Division Polynomials and Somos Sequences

In this paper we consider the sequences $(Ψ_{n}(\mathbf{P}))_{n\geq 0}$, $(Φ_{n}(\mathbf{P}))_{n\geq 0}$ and $(\overline{Ω}\,_{n}(\mathbf{P}% ))_{n\geq 0}$ of values of the translated division polynomials of an elliptic curve $E/K$ evaluated at a point $\mathbf{P}\in $ $E(K)^{2}$. We prove that these sequences are purely periodic when $K$ is a finite field. Then we use the periodicity properties of these sequences to prove that certain subsequences of these sequences are $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion _{p}$-Cauchy. Finally, we use this result to prove analogous results for Somos $4$ and Somos $5$ sequences.

math.NT↗

arXiv2019

Sequences associated to elliptic curves

Let $E$ be an elliptic curve defined over a field $K$ (with $char(K)\neq 2$) given by a Weierstrass equation and let $P=(x,y)\in E(K)$ be a point. Then for each $n$ $\geq 1$ and some $γ\in K^{\ast }$ we can write the $x$- and $y$-coordinates of the point $[n]P$ as \begin{equation*} \lbrack n]P=\left( \frac{ϕ_n(P)}{ψ_n^2(P)}, \frac{ω_n(P) }{ψ_n^3(P)} \right) =\left( \frac{γ^2 G_n(P)}{F_n^2(P)}, \frac{γ^3 H_n(P)}{F_n^3(P)}\right) \end{equation*} where $ϕ_n,ψ_n,ω_n \in K[x,y]$, $\gcd (ϕ_n,ψ_n^2)=1$ and \begin{equation*} F_n(P) = γ^{1-n^2}ψ_n(P), G_n(P) = γ^{-2n^{2}}ϕ_{n}(P),H_{n}(P) = γ^{-3n^{2}} ω_n(P) \end{equation*} are suitably normalized division polynomials of $E$. In this work we show the coefficients of the elliptic curve $E$ can be defined in terms of the sequences of values $(G_{n}(P))_{n\geq 0}$ and $(H_{n}(P))_{n\geq 0}$ of the suitably normalized division polynomials of $E$ evaluated at a point $P \in E(K)$. Then we give the general terms of the sequences $(G_{n}(P))_{n\geq 0}$ and $(H_{n}(P))_{n\geq 0}$ associated to Tate normal form of an elliptic curve. As an application of this we determine square and cube terms in these sequences.

math.NT↗

arXiv2011

Elliptic Divisibility Sequences, Squares and Cubes

Elliptic divisibility sequences (EDSs) are generalizations of a class of integer divisibility sequences called Lucas sequences. There has been much interest in cases where the terms of Lucas sequences are squares or cubes. In this work, using the Tate normal form having one parameter of elliptic curves with torsion points, the general terms and periods of all elliptic divisibility sequences with a zero term are given in terms of this parameter by means of Mazur's theorem, and it is shown that which term of h_{n} of an EDS can be a square or a cube by using the general terms of these sequences.

math.NT↗