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

Abstract

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 new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group.

Torsors for finite group schemes of bounded height

“…1.2.1. Our first motivation comes from recent attempts [17,21] to extend a conjecture of Manin on the number of rational points of bounded height on Fano varieties [23] to algebraic stacks. The articles try to explain à la Manin some counting results on points of stacks (such as [30]), as well as similarities between the Manin and the Malle conjectures, observed by the second named author in [53].…”

“…For that purpose, definitions of heights for stacks are provided. The generalization of the Manin conjecture for stacks has not been completed: in [17] one is restricted to weighted projective stacks , while in [21], the authors predict the exponent of $$B$$ up to an $$\epsilon$$ in an asymptotic formula on the number of points, but no leading constant or the exponent of $$\log (B)$$. The article [18] is concerned with this question.…”

Stringy invariants and toric Artin stacks

“…leading to a flurry of activity in the subject [BG09, CCIT09, CCIT15, CIJ18]. Twisted curves have even seen applications in number theory, where the first author, Ellenberg, and Zureick-Brown [ESZB23] generalized the theory of Weil heights to the case of stacks and gave point-counting heuristics which unified the Batyrev-Manin and Malle conjectures. Inspired by [AV02], Yasuda [Yas04] introduced the moduli space of twisted arcs.…”

“…The definition of height function we introduced was inspired by[ESZB23] where the numbertheoretic Weil height functions were extended to the case of vector bundles on stacks.…”

“…This definition was inspired by recent work [ESZB23] of the first author, Ellenberg, and Zureick-Brown, where we generalized the notion of number-theoretic Weil heights to the case of stacks, and unified the Batyrev-Manin and Malle Conjectures.…”