加权投射堆中的薄集

Motive of the moduli stack of rational curves on a weighted projective stack

We show the compactly supported motive of the moduli stack of degree $n$ rational curves on the weighted projective stack $\mathcal{P}(a,b)$ is of mixed Tate type over any base field $K$ with $\text{char}(K) \nmid a,b$ and has class $\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}$ in the Grothendieck ring of stacks. In particular, this improves upon the result of [HP] regarding the arithmetic invariant of the moduli stack $\mathcal{L}_{1,12n} := \mathrm{Hom}_{n}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$ of stable elliptic fibrations over $\mathbb{P}^{1}$ with $12n$ nodal singular fibers and a …

Rational points of bounded height on weighted projective stacks

A weighted projective stack is a stacky quotient $\mathscr P(\mathbf a)=(\mathbf A^n-\{0\})/\mathbb G_m$, where the action of $\mathbb G_m$ is with weights $\mathbf a\in\mathbb Z^n_{>0}$. Examples are: the compactified moduli stack of elliptic curves $\mathscr P(4,6)$ and the classifying stack of $μ_m$-torsors $Bμ_m=\mathscr P(m)$. We define heights on the weighted projective stacks. The heights generalize the naive height of an elliptic curve and the absolute discriminant of a torsor. We use the heights to count rational points. We find the asymptotic behaviour for the number of rational poin…

有界高度点在加权投射堆映射中的应用及椭圆曲线计数

Points of bounded height in images of morphisms of weighted projective stacks with applications to counting elliptic curves

研究有界高度点在加权投射堆映射中的渐进行为，并应用于计数具有指定水平结构的椭圆曲线，包括Γ(N)、Γ₁(N)和Γ₀(N)等情形。

Heights on stacks and a generalized Batyrev-Manin-Malle conjecture

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something n…

整体函数域上加权射影空间中的有界高度点

Points of bounded height on weighted projective spaces over global function fields

本文研究整体函数域上加权射影叠的有界高度有理点计数问题，给出了点数的精确公式与渐近式。

Étale cohomological stability of the moduli space of stable elliptic surfaces

We compute the (stable) étale cohomology of $\mathrm{Hom}_{n}(C, \mathcal{P}(\vecλ))$, the moduli stack of degree $n$ morphisms from a smooth projective curve $C$ to the weighted projective stack $\mathcal{P}(\vecλ)$, the latter being a stacky quotient defined by $\mathcal{P}(\vecλ) := \left[\mathbb{A}^N-\{0\}/\mathbb{G}_m\right]$, where $\mathbb{G}_m$ acts by weights $\vecλ = (λ_0, \cdots, λ_N) \in \mathbb{Z}^N_{+}$. Our key ingredient is formulating and proving the étale cohomological descent over the category $ΔS$, the symmetric (semi)simplicial category. An immediate arithmetic consequence…