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PreprintCC BY 4.0arXiv

Counterexamples to O'Neil's Period-Index Problem on Elliptic Curves

  • Xiaoguang Shang
  • Cheng Niu
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Abstract

Let E/K be an elliptic curve over a number field K, and let [C]∈ H1(K,E(K̅)) be a homogeneous space under E. Suppose that C has period n and index d. O'Neil asked whether one can always choose a lift of [C] to H1(K,E[n]) whose period-index obstruction has order exactly d/n. We give a negative answer to this question by constructing an explicit family of examples.Taking K=ℚ(ζ8), we prove that there exist infinitely many pairwise non-isomorphic elliptic curves E/K such that each E admits infinitely many homogeneous spaces [C]∈ H1(K,E(K̅)) with period $8$ and index $16$, but the period-index obstruction of every lift of [C] to H1(K,E[8]) has order exactly $8$.

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Cite

BibTeXRIS
BibTeX
@misc{shang2026counterexamples,
  title = {{Counterexamples to O'Neil's Period-Index Problem on Elliptic Curves}},
  author = {Xiaoguang Shang and Cheng Niu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29288},
  eprint = {2608.29288},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29288},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Counterexamples to O'Neil's Period-Index Problem on Elliptic Curves
AU  - Xiaoguang Shang
AU  - Cheng Niu
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29288
UR  - https://arxiv.org/abs/2608.29288
AB  - Let E/K be an elliptic curve over a number field K, and let [C]∈ H^1(K,E(K̅)) be a homogeneous space under E. Suppose that C has period n and index d. O'Neil asked whether one can always choose a lift of [C] to H^1(K,E[n]) whose period-index obstruction has order exactly d/n. We give a negative answer to this question by constructing an explicit family of examples.Taking K=ℚ(ζ_8), we prove that there exist infinitely many pairwise non-isomorphic elliptic curves E/K such that each E admits infinitely many homogeneous spaces [C]∈ H^1(K,E(K̅)) with period $8$ and index $16$, but the period-index obstruction of every lift of [C] to H^1(K,E[8]) has order exactly $8$.
ER  -