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arXiv · 2607.11261

p-adic Properties of Translated Division Polynomials and Somos Sequences

Betül Gezer·Osman Bizim
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Abstract

In this paper we consider the sequences $(\Psi _{n}(\mathbf{P}))_{n\geq 0}$, $(\Phi _{n}(\mathbf{P}))_{n\geq 0}$ and $(\overline{\Omega }\,_{n}(\mathbf{P}% ))_{n\geq 0}$ 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.

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Betül Gezer, Osman Bizim. 2026-07-13. p-adic Properties of Translated Division Polynomials and Somos Sequences. https://arxiv.org/abs/2607.11261

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