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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Global Weierstrass equations of hyperelliptic curves
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by Qing Liu PDF
Trans. Amer. Math. Soc. 375 (2022), 5889-5906 Request permission


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:
  • 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:
  • MathSciNet review: 4469240