Geodesic
Heights on stacks and a generalized Batyrev–Manin–Malle conjecture
Forum of Mathematics, Sigma, Tome 11 (2023)

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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. 
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     author = {Jordan S. Ellenberg and Matthew Satriano and David Zureick-Brown},
     title = {Heights on stacks and a generalized {Batyrev{\textendash}Manin{\textendash}Malle} conjecture},
     journal = {Forum of Mathematics, Sigma},
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Jordan S. Ellenberg; Matthew Satriano; David Zureick-Brown. Heights on stacks and a generalized Batyrev–Manin–Malle conjecture. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.5

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