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PreprintarXiv licencearXiv

Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves

  • Martí Oller
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Abstract

We determine asymptotic results for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus g≥ 2. Our results come from two different routes. The first route is through the geometry-of-numbers methods pioneered by Bhargava, where we obtain new parametrisations coming from Vinberg theory arising from representations related to the Dynkin diagrams of type B and C. The second route uses recent results by Koymans--Smith, which relate the average sizes of Selmer groups to Tamagawa ratios through a formula by Greenberg--Wiles.

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Cite

BibTeXRIS
BibTeX
@misc{oller2026arithmetic,
  title = {{Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves}},
  author = {Martí Oller},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.06327},
  eprint = {2606.06327},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.06327},
}
RIS
TY  - UNPB
TI  - Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves
AU  - Martí Oller
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2606.06327
UR  - https://arxiv.org/abs/2606.06327
AB  - We determine asymptotic results for the average size of Selmer groups arising from certain isogenies related to Jacobians of hyperelliptic curves of genus g≥ 2. Our results come from two different routes. The first route is through the geometry-of-numbers methods pioneered by Bhargava, where we obtain new parametrisations coming from Vinberg theory arising from representations related to the Dynkin diagrams of type B and C. The second route uses recent results by Koymans--Smith, which relate the average sizes of Selmer groups to Tamagawa ratios through a formula by Greenberg--Wiles.
ER  -