论文标题
高度模量在循环堆栈上的高度模量和在功能场上计数椭圆曲线
Height moduli on cyclotomic stacks and counting elliptic curves over function fields
论文作者
论文摘要
对于适当的堆栈，与方案不同，有理点和积分点之间存在区别。此外，理性点具有额外的自动形态群体。我们表明，这些区别准确地说明了较低的主要术语，以椭圆曲线的精确计数在功能字段上出现，在这种情况下回答了Venkatesh的问题。更普遍地，使用Ellenberg，Zureick-Brown和第三作者最近引入的扭曲稳定地图和堆叠高度函数的理论，我们构建了有限类型的模量空间，该空间参数可以在一大批堆栈，所谓的环测细胞堆栈上参数固定高度的理性点。主要工具是理性点，扭曲地图和加权线性系列之间的对应关系。在此过程中，我们获得了诺斯科特（Northcott）的特性以及泰特（Tate）的算法的概括，用于循环堆栈，并计算这些模量空间的精确动机，以进行加权射击堆栈。
For proper stacks, unlike schemes, there is a distinction between rational and integral points. Moreover, rational points have extra automorphism groups. We show that these distinctions exactly account for the lower order main terms appearing in precise counts of elliptic curves over function fields, answering a question of Venkatesh in this case. More generally, using the theory of twisted stable maps and the stacky height functions recently introduced by Ellenberg, Zureick-Brown, and the third author, we construct finite type moduli spaces which parametrize rational points of fixed height on a large class of stacks, so-called cyclotomic stacks. The main tool is a correspondence between rational points, twisted maps and weighted linear series. Along the way, we obtain the Northcott property as well as a generalization of Tate's algorithm for cyclotomic stacks, and compute the exact motives of these moduli spaces for weighted projective stacks.
