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 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.

Stacky heights on elliptic curves in characteristic 3

“…In this paper, we investigate heights on the compactified moduli stack of elliptic curves in characteristic 3. We show that the notion of stacky height introduced in [ESZB21] does not always recover the classical notion of height. Specifically, we show there is no vector bundle whose associated stacky height induces the usual notion of Faltings height for elliptic curves in characteristic 3.…”

“…That is, ω := f * ω E /(M 1,1 ) k ′ for f : E → (M 1,1 ) k ′ the universal stable elliptic curve. Then [ESZB21,Proposition 3.11] show that, for K a finite extension of k ′ (t), and x : Spec K → (M 1,1 ) k ′ a point, ht ω (x) = ht(x), with the latter notion of Faltings height as defined in Definition 1.1. However, as [ESZB21,p.…”

“…Then [ESZB21,Proposition 3.11] show that, for K a finite extension of k ′ (t), and x : Spec K → (M 1,1 ) k ′ a point, ht ω (x) = ht(x), with the latter notion of Faltings height as defined in Definition 1.1. However, as [ESZB21,p. 27] observe, for k a field of characteristic 3, it is no longer true that ht ω (x) = ht(x) for all x : Spec K → M 1,1 .…”

“…In particular, they show cubic twists of the form y 2 = x 3 − x + f (t), for f (t) ∈ k[t], all have height 0 with respect to the Hodge bundle, even though their Faltings heights can be nonzero. Moreover, they show there is no line bundle L on M 1,1 for which ht L (x) = ht(x) [ESZB21,p. 27].…”

“…We say a height function on the set of k(t) points of a stack X satisfies the Northcott property if there are finitely many such points of bounded height, cf. [ESZB21,p. 4].…”