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- W4385964586 abstract "In this article we consider smooth projective curves $C$ of genus two described by integral equations of the form $y^2=xh(x)$, where $h(x)inmathbb{Z}[x]$ is monic of degree $4$. It turns out that if $h(x)$ is reducible, then the absolute discriminant of $C$ can never be an odd prime, except when $h(x)=(x-b)g(x)$ and $g(x)$ is irreducible. In this case we obtain a complete description of such genus $2$ curves. In fact, we prove that there are two one-parameter families $C_t^i$, $i=1,2$, of such curves such that if $C$ is a genus two curve with an odd prime absolute discriminant, then $C$ is $C_t^i$, for some $i$, and $tinmathbb{Z}$. Moreover, we show that $C_t^i$ has an odd prime absolute discriminant, $p$, if and only if a certain degree-$4$ irreducible polynomial $f^i(t)inmathbb{Z}[t]$ takes the value $p$ at $t$. Hence there are conjecturally infinitely many such curves. When $h(x)$ is irreducible, we give explicit examples of one-parameter families of genus $2$ curves $C_t$ such that $C_t$ has an odd prime absolute discriminant for conjecturally infinitely many integer values $t$." @default.
- W4385964586 created "2023-08-18" @default.
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- W4385964586 date "2020-03-19" @default.
- W4385964586 modified "2023-09-23" @default.
- W4385964586 title "Genus 2 curves with bad reduction at one odd prime" @default.
- W4385964586 doi "https://doi.org/10.48550/arxiv.2003.09010" @default.
- W4385964586 hasPublicationYear "2020" @default.
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