## Global Weierstrass equations of hyperelliptic curves

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- Trans. Amer. Math. Soc.
**375**(2022), 5889-5906 Request permission

## Abstract:

Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $\mathscr {C}$ of $C$ over the ring of integers $\mathcal {O}_K$ (*i.e.*the hyperelliptic involution of $C$ extends to $\mathscr {C}$ and the quotient is a smooth model of $\mathbb {P}^1_K$ over $\mathcal {O}_K$), we give necessary and sometimes sufficient conditions for $\mathscr {C}$ to be defined by a global Weierstrass equation. In particular, if $C$ has everywhere good reduction, we prove that it is defined by a global integral Weierstrass equation with invertible discriminant if the class number $h_K$ is prime to $2(2g+1)$, confirming a conjecture of M. Sadek.

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## Additional Information

**Qing Liu**- Affiliation: Université de Bordeaux, Institut de Mathématiques de Bordeaux, CNRS UMR 5251, 33405 Talence, France
- MR Author ID: 240790
- ORCID: 0000-0001-6884-139X
- Email: Qing.Liu@math.u-bordeaux.fr
- Received by editor(s): July 9, 2021
- Received by editor(s) in revised form: February 3, 2022, February 6, 2022, and February 16, 2022
- Published electronically: June 3, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5889-5906 - MSC (2020): Primary 11G30, 11G05, 14D10, 14H25
- DOI: https://doi.org/10.1090/tran/8672
- MathSciNet review: 4469240