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PreprintCC BY-NC-ND 4.0arXiv

p-adic Properties of Translated Division Polynomials and Somos Sequences

  • Betül Gezer
  • Osman Bizim
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Abstract

We study the periodicity and p-adic convergence of sequences of values of the translated division polynomials (Ψn(P))n≥ 0, (Φn(P))n≥ 0, and (Ωn(P))n≥ 0 evaluated at a pair of points on an elliptic curve. We prove that these sequences are purely periodic over finite fields. We then use their periodicity properties to prove that certain subsequences are ℤp-Cauchy. Applying this result to Somos 4 and Somos 5 sequences yields analogous convergence results. We also establish pure periodicity modulo prime powers for the even- and odd-indexed subsequences of Somos 5 sequences under suitable hypotheses.

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Cite

BibTeXRIS
BibTeX
@misc{gezer2026padic,
  title = {{p-adic Properties of Translated Division Polynomials and Somos Sequences}},
  author = {Betül Gezer and Osman Bizim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.11261},
  eprint = {2607.11261},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.11261},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - p-adic Properties of Translated Division Polynomials and Somos Sequences
AU  - Betül Gezer
AU  - Osman Bizim
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2607.11261
UR  - https://arxiv.org/abs/2607.11261
AB  - We study the periodicity and p-adic convergence of sequences of values of the translated division polynomials (Ψ_n(P))_n≥ 0, (Φ_n(P))_n≥ 0, and (Ω_n(P))_n≥ 0 evaluated at a pair of points on an elliptic curve. We prove that these sequences are purely periodic over finite fields. We then use their periodicity properties to prove that certain subsequences are ℤ_p-Cauchy. Applying this result to Somos 4 and Somos 5 sequences yields analogous convergence results. We also establish pure periodicity modulo prime powers for the even- and odd-indexed subsequences of Somos 5 sequences under suitable hypotheses.
ER  -