p-adic Properties of Translated Division Polynomials and Somos Sequences

Abstract

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.