Ekedahl-Oort strata of hyperelliptic curves in characteristic 2
Ekedahl-Oort strata of hyperelliptic curves in characteristic 2
复制标题
特征 2 中超椭圆曲线的 Ekedahl-Oort 地层
DOI：
10.2140/ant.2013.7.507
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
R. Pries
中科院分区：
文献类型：
--
作者：
Arsen Elkin;R. Pries
设X是定义在特征为p=2的代数闭域k上的亏格为g的超椭圆曲线。我们证明了$X$的de Rham上同调分解成由超椭圆覆盖的分支点索引的片段。这使得我们可以计算$X$的Jacobian的$2$-挠群方案$J_X[2]$的Ekedahl-Oort型同构类。有趣的是，$J_X[2]$仅依赖于$X$的某些离散不变量，即依赖于与分支点相关的分歧不变量。给出了任意亏格的超椭圆k-曲线的Jacobian的2-扭群格式的完全分类。
Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p=2$. We prove that the de Rham cohomology of $X$ decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the $2$-torsion group scheme $J_X[2]$ of the Jacobian of $X$ in terms of the Ekedahl-Oort type. The interesting feature is that $J_X[2]$ depends only on some discrete invariants of $X$, namely, on the ramification invariants associated with the branch points. We give a complete classification of the group schemes which occur as the $2$-torsion group schemes of Jacobians of hyperelliptic $k$-curves of arbitrary genus.