论文标题
探索费马特类型的阿伯利亚品种中的堕落性
An Exploration of Degeneracy in Abelian Varieties of Fermat Type
论文作者
论文摘要
该术语简并用来描述其hodge环包含特殊循环的阿贝尔品种 - 霍奇循环，这些周期不是由分层类别产生的。我们可以通过其Mumford-Tate组，Hodge Group和Sato-Tate组看到特殊周期对Abelian品种结构的影响。在本文中，我们通过这些不同但相关的镜头来研究退化。我们专注于一个阿贝尔（Abelian）类型的阿伯里亚（Abelian）品种，即形式的jacobians $ y^2 = x^m-1 $的高ellip曲曲线。我们证明，每当$ m $是一个奇怪的复合整数时，曲线的雅各比式就会退化。我们探索了几个例子的各种堕落性，每个示例都说明了可能发生的不同现象。
The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that the Jacobian of the curve is degenerate whenever $m$ is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.
