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PreprintarXiv licencearXiv

Likely and unlikely ramification of algebraic points on curves

  • Bianca Viray
  • Isabel Vogt
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Abstract

This paper initiates the study of arithmetic properties of closed points of arbitrary degree on curves. We show that a pointless curve of rank $0$ can have only finitely many cyclic degree ℓ points for all sufficiently large primes ℓ. More generally, we show that the residue fields of points on curves of genus at least $2$ and rank $0$ are often biased away from total ramification. The proof technique illuminates a natural intermediate conjecture between the Manin--Mumford conjecture and the Zilber--Pink conjecture for self-products of abelian varieties. This intermediate conjecture implies analogous results for other splitting profiles.

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Cite

BibTeXRIS
BibTeX
@misc{viray2026likely,
  title = {{Likely and unlikely ramification of algebraic points on curves}},
  author = {Bianca Viray and Isabel Vogt},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17262},
  eprint = {2610.17262},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17262},
}
RIS
TY  - UNPB
TI  - Likely and unlikely ramification of algebraic points on curves
AU  - Bianca Viray
AU  - Isabel Vogt
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17262
UR  - https://arxiv.org/abs/2610.17262
AB  - This paper initiates the study of arithmetic properties of closed points of arbitrary degree on curves. We show that a pointless curve of rank $0$ can have only finitely many cyclic degree ℓ points for all sufficiently large primes ℓ. More generally, we show that the residue fields of points on curves of genus at least $2$ and rank $0$ are often biased away from total ramification. The proof technique illuminates a natural intermediate conjecture between the Manin--Mumford conjecture and the Zilber--Pink conjecture for self-products of abelian varieties. This intermediate conjecture implies analogous results for other splitting profiles.
ER  -