论文标题
有限高度的有限组方案
Torsors for finite group schemes of bounded height
论文作者
论文摘要
令$ f $为全球领域。令$ g $为非琐碎的有限étaletame $ f $ group方案。我们在$ f上定义了$ g $ -torsors的集合的高度功能，$ $ $ $ $ $ $ $ $。作为对小组计划的Malle猜想的类似物，我们对$ g $ -torsors数量的渐近行为提出了一个猜想。这是我们从Arxiv：2207.03645的更为普遍的堆叠Batyrev-Manin猜想的特殊情况。对于$ g $的案件证明了渐近造型的渐近性。当$ f $是一个数字字段时，引导常数表示为某些算术不变的$ g $的产物，以及附加到$ g $的空间的体积。此外，在该空间中建立了$ g $ torsors的等分分配属性。
Let $F$ be a global field. Let $G$ be a non trivial finite étale tame $F$-group scheme. We define height functions on the set of $G$-torsors over $F,$ which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of $G$-torsors over $F$ of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case $G$ is commutative. When $F$ is a number field, the leading constant is expressed as a product of certain arithmetic invariants of $G$ and a volume of a space attached to $G$. Moreover, an equidistribution property of $G$-torsors in the space is established.
