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PreprintCC BY 4.0arXiv

p-adic properties of division polynomials and algebraic sigma functions

  • Yu Katagiri
  • Shinichi Kobayashi
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Abstract

Let p ≥ 5 be a prime, let K be a finite extension of ℚp, and let E/K be an elliptic curve with good reduction. Let Fn denote the n-division polynomial of E. Silverman proved that if the reduction is ordinary, then for every P ∈ E(K) ∖ Ê(K) and a suitable power q of p, the sequences (Fmqk(P))k ≥ 0 converge p-adically to limits that are algebraic over the field of definition of E. In both the ordinary and supersingular cases, we show that these sequences converge, and determine these limits explicitly in terms of the values of Mumford's algebraic sigma function attached to the Teichmüller lift of the prime-to-p torsion component of the reduction of P. In particular, the limits are algebraic in the supersingular case as well. As an application, we obtain explicit p-adic limit formulas for nonsingular elliptic divisibility sequences.

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Cite

BibTeXRIS
BibTeX
@misc{katagiri2026padic,
  title = {{$p$-adic properties of division polynomials and algebraic sigma functions}},
  author = {Yu Katagiri and Shinichi Kobayashi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07024},
  eprint = {2609.07024},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07024},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - p-adic properties of division polynomials and algebraic sigma functions
AU  - Yu Katagiri
AU  - Shinichi Kobayashi
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07024
UR  - https://arxiv.org/abs/2609.07024
AB  - Let p ≥ 5 be a prime, let K be a finite extension of ℚ_p, and let E/K be an elliptic curve with good reduction. Let F_n denote the n-division polynomial of E. Silverman proved that if the reduction is ordinary, then for every P ∈ E(K) ∖ Ê(K) and a suitable power q of p, the sequences (F_mq^k(P))_k ≥ 0 converge p-adically to limits that are algebraic over the field of definition of E. In both the ordinary and supersingular cases, we show that these sequences converge, and determine these limits explicitly in terms of the values of Mumford's algebraic sigma function attached to the Teichmüller lift of the prime-to-p torsion component of the reduction of P. In particular, the limits are algebraic in the supersingular case as well. As an application, we obtain explicit p-adic limit formulas for nonsingular elliptic divisibility sequences.
ER  -