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GENUS
$2$ CURVES WITH BAD REDUCTION AT ONE ODD PRIME
- Part of
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- Journal:
- Nagoya Mathematical Journal / Volume 254 / June 2024
- Published online by Cambridge University Press:
- 29 November 2023, pp. 498-512
- Print publication:
- June 2024
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- Article
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The problem of classifying elliptic curves over
$\mathbb Q$ with a given discriminant has received much attention. The analogous problem for genus 
$2$ curves has only been tackled when the absolute discriminant is a power of 
$2$. In this article, we classify genus 
$2$ curves C defined over 
${\mathbb Q}$ with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many 
$1$-parameter families of genus 
$2$ curves. In particular, we provide genus 
$2$ analogues to Neumann–Setzer families of elliptic curves over the rationals.
