论文标题
属2曲线曲线，一个奇数质量降低不良
Genus 2 curves with bad reduction at one odd prime
论文作者
论文摘要
在本文中，我们考虑了平滑的投射曲线$ c $属的属属，由$ y^2 = xh（x）$的积分方程描述，其中$ h（x）\ in \ mathbb {z} [x] $是$ 4 $的一元人。事实证明，如果$ h（x）$可还原，那么$ c $的绝对判别物永远不会是一个奇怪的素数，除非$ h（x）=（x-b）g（x）$和$ g（x）$是不可记录的。在这种情况下，我们获得了此类$ 2 $曲线的完整描述。实际上，我们证明有两个单参数系列$ c_t^i $，$ i = 1,2 $的曲线，因此，如果$ c $是具有奇怪质量绝对判别的属属的两曲线，那么$ c $ as $ c_t^i $，对于某些$ i $，以及$ i $，以及$ t \ in \ mathbb in \ mathbb {z z} $。此外，我们表明$ c_t^i $具有奇怪的绝对判别，$ p $，并且只有一定程度 - $ 4 $ $ 4 $不可差的多项式$ f^i（t）\ in \ mathbb {z} [z} [t] $将$ p $ p $ at $ t $。因此，有许多这样的曲线无限地构想。当$ h（x）$是不可约的时，我们给出了一个参数属属$ 2 $ curves $ c_t $的明确示例，使得$ c_t $具有奇怪的质量绝对绝对歧视，因为他们自以为是无限的整数值$ t $。
In this article we consider smooth projective curves $C$ of genus two described by integral equations of the form $y^2=xh(x)$, where $h(x)\in\mathbb{Z}[x]$ 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 $t\in\mathbb{Z}$. Moreover, we show that $C_t^i$ has an odd prime absolute discriminant, $p$, if and only if a certain degree-$4$ irreducible polynomial $f^i(t)\in\mathbb{Z}[t]$ 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 $t$.
