[Paper Review] Heights on stacks and a generalized Batyrev-Manin-Malle conjecture

Jordan S. Ellenberg, Matthew Satriano+1 more|arXiv (Cornell University)|Jun 21, 2021

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper introduces a new notion of height for rational points on algebraic stacks over global fields, generalizing both the Batyrev–Manin conjecture for rational points on varieties and Malle’s conjecture for number fields. The authors define heights using vector bundles on stacks, compute them for key examples like classifying stacks, symmetric powers, and weighted projective stacks, and formulate a unified conjecture predicting asymptotic growth of rational points of bounded height.

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 X, which specializes to the Baytev-Manin conjecture when X is a scheme and to Malle's conjecture when X is the classifying stack of a finite group.

Motivation & Objective

• To address the lack of a standard height definition for rational points on algebraic stacks, which are not embeddable in projective space like schemes.
• To generalize the classical Batyrev–Manin conjecture on rational points of bounded height on Fano varieties and Malle’s conjecture on number fields with bounded discriminant.
• To provide a uniform framework for measuring the complexity of rational points on stacks using vector bundles and their associated heights.
• To compute the new height invariant for various important stacks, including $BG$, symmetric powers of varieties, moduli stacks of abelian varieties, and weighted projective stacks.
• To formulate a conjecture for the asymptotic count of rational points of bounded height on a stack, which specializes to both the Batyrev–Manin and Malle conjectures in appropriate cases.

Proposed method

• Define a height function on rational points of a proper algebraic stack with finite diagonal over a global field using a vector bundle, generalizing Weil’s height on projective varieties.
• Use the notion of a 'tuning stack' and 'tuning sheaf' to construct a height via pullback to a finite cover of the stack by a scheme.
• Introduce a local discrepancy correction to ensure consistency in the height definition across different covers and stacky structures.
• Apply the height definition to specific stacks: $BG$ for finite groups, $B\mu_n$, symmetric powers of varieties, weighted projective stacks, and moduli stacks of abelian varieties.
• Use Arakelov theory and intersection theory on stacky curves to define the degree of line bundles and torsion sheaves via finite flat covers by regular schemes.
• Formulate a conjecture on the asymptotic growth of the number of rational points of bounded height on a stack, with exponents determined by a 'stacky anti-canonical height' and expected deformation dimension.

Experimental results

Research questions

• RQ1 How can one define a height invariant for rational points on algebraic stacks that generalizes both the classical height on varieties and the discriminant-based counting in number fields?
• RQ2 In what cases does the proposed height construction agree with existing ad hoc notions of complexity in arithmetic geometry?
• RQ3 What is the asymptotic behavior of the number of rational points of bounded height on a stack, and how does it unify the Batyrev–Manin and Malle conjectures?
• RQ4 How does the height behave under pullbacks and finite maps in the context of stacky curves and higher-dimensional stacks?
• RQ5 What role does the expected deformation dimension (edd) play in predicting the growth rate of rational points on stacks?

Key findings

• The proposed height construction generalizes the classical Weil height on projective varieties and recovers known height notions on $BG$ and weighted projective stacks.
• For the classifying stack $BG$ of a finite group $G$, the height function corresponds to the discriminant of a $G$-extension, recovering Malle’s conjecture in the asymptotic count of number fields.
• For symmetric powers of $\mathbb{P}^n$, the height function aligns with standard height definitions, and the conjecture predicts asymptotic growth of the form $cB^{a}(\log B)^b$ with explicit exponents.
• On weighted projective stacks, the height function matches existing definitions by Deng and Beshaj–Gutierrez–Shaska, validating the construction in known cases.
• The conjecture predicts that when the expected deformation dimension (edd) is negative, the number of rational points of bounded height is finite, suggesting a stacky analogue of the Lang–Vojta conjecture.
• The height is well-defined and independent of the choice of finite flat cover via the degree normalization in Definition B.5, ensuring consistency in arithmetic intersection theory on stacks.