论文标题
DM堆栈的Batyrev-Manin猜想
The Batyrev-Manin conjecture for DM stacks
论文作者
论文摘要
我们在一个数字字段上的DM（Deligne-Mumford）堆栈的合理点上定义了一个新的高度功能。这概括了Ellenberg-venkatesh的广义判别，这是Ellenberg-Satriano-Zureick-Brown最近引入的高度函数（就数字字段而言，就DM堆栈而言），以及darda对加权投影堆栈的准高度高度函数。概括了Manin的猜想和更一般的Batyrev-Manin猜想，我们就具有有界高度的DM堆栈的理性点数量的渐近行为进行了一些猜想。为了制定DM堆栈的Batyrev-manin猜想，我们介绍了所谓的$ a $ a $ a-和$ b $ invariants的Orbifold版本。当应用于有限组的分类堆栈时，这些猜想专门针对Malle的猜想，只是我们从计数中删除了某些薄色子集。更确切地说，我们删除了包括Hassett，Tschinkel，Tanimoto，Tanimoto，Lehmann和Sengupta在内的品种中研究的破碎薄亚集，并且由于我们对$ a $ a $ a $ a $ a - 和$ b $ invariants的概括，可以将其推广到DM堆栈。破裂的薄子集使我们能够重新解释Klünern的反例，以示例对Malle猜想。
We define a new height function on rational points of a DM (Deligne-Mumford) stack over a number field. This generalizes a generalized discriminant of Ellenberg-Venkatesh, the height function recently introduced by Ellenberg-Satriano-Zureick-Brown (as far as DM stacks over number fields are concerned), and the quasi-toric height function on weighted projective stacks by Darda. Generalizing the Manin conjecture and the more general Batyrev-Manin conjecture, we formulate a few conjectures on the asymptotic behavior of the number of rational points of a DM stack with bounded height. To formulate the Batyrev-Manin conjecture for DM stacks, we introduce the orbifold versions of the so-called $a$- and $b$-invariants. When applied to the classifying stack of a finite group, these conjectures specialize to the Malle conjecture, except that we remove certain thin subsets from counting. More precisely, we remove breaking thin subsets, which have been studied in the case of varieties by people including Hassett, Tschinkel, Tanimoto, Lehmann and Sengupta, and can be generalized to DM stack thanks to our generalization of $a$- and $b$-invariants. The breaking thin subset enables us to reinterpret Klüners' counterexample to the Malle conjecture.
