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PreprintCC BY-NC-ND 4.0arXiv

Factorization of Elliptic Divisibility Sequences via Modular Curves

  • Betül Gezer
  • İpek Çolak
  • Osman Bizim
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Abstract

In this paper, we give a structural factorization of elliptic divisibility sequences in terms of finitely many modular functions on the modular curve X1(N). More precisely, we express the functions obtained by pulling back division polynomials along the universal torsion section as products of fixed modular factors. Then we show that this factorization is compatible with the natural action of (ℤ/Nℤ)× on the modular curve X1(N) and so descends to the modular curve X0(N) via the norm map. As applications, we obtain a perfect power criterion, local and global growth estimates for the associated elliptic divisibility sequences.

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Cite

BibTeXRIS
BibTeX
@misc{gezer2026factorization,
  title = {{Factorization of Elliptic Divisibility Sequences via Modular Curves}},
  author = {Betül Gezer and İpek Çolak and Osman Bizim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37191},
  eprint = {2610.37191},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37191},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Factorization of Elliptic Divisibility Sequences via Modular Curves
AU  - Betül Gezer
AU  - İpek Çolak
AU  - Osman Bizim
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37191
UR  - https://arxiv.org/abs/2610.37191
AB  - In this paper, we give a structural factorization of elliptic divisibility sequences in terms of finitely many modular functions on the modular curve X_1(N). More precisely, we express the functions obtained by pulling back division polynomials along the universal torsion section as products of fixed modular factors. Then we show that this factorization is compatible with the natural action of (ℤ/Nℤ)^× on the modular curve X_1(N) and so descends to the modular curve X_0(N) via the norm map. As applications, we obtain a perfect power criterion, local and global growth estimates for the associated elliptic divisibility sequences.
ER  -