Global Weierstrass equations of hyperelliptic curves

Liu, Qing

doi:10.1090/tran/8672

Global Weierstrass equations of hyperelliptic curves
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by Qing Liu PDF

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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