Abstract
We describe an algorithm for determining a minimal Weierstrass equation for hyperelliptic curves over principal ideal domains. When the curve has a rational Weierstrass point \(w_0\), we also give a similar algorithm for determining the minimal Weierstrass equation with respect to \(w_0\).
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Notes
Let \(g\ge 1\) be odd and let \(p>2\). Consider the equation \(y^2=px^{2g+1}+p^{g+2}\) over \({\mathbb {Z}}_p\). Then \(\epsilon =1\). For the point \(x=y=p=0\), we have \(\lambda =g+2\). By (2.b), this equation is minimal. But \(\lambda >g+1\). So in (1) the converse does not hold for odd g in general.
References
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The PARI Group, PARI/GP, Univ. Bordeaux, 2023, available from http://pari.math.u-bordeaux.fr/
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I would like to thank Bill Allombert for clarifications regarding some computational aspects in this article and for pointing out related references. I would also like to thank the referees for their thorough reading. Thank you also to the referees and Bill Allombert for suggestions which led to improvements in the presentation of this manuscript.
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