Height moduli on cyclotomic stacks and counting elliptic curves over function fields

TL;DR

This paper explores the counting of elliptic curves over function fields through the lens of cyclotomic stacks, revealing how stack-specific properties influence point counts and establishing new moduli space constructions.

Contribution

It introduces a framework linking rational points on cyclotomic stacks with twisted maps and linear series, addressing a question by Venkatesh and generalizing Tate's algorithm.

Findings

Lower order main terms in elliptic curve counts explained by stack distinctions.

Constructed finite type moduli spaces for rational points of fixed height on cyclotomic stacks.

Established the Northcott property and computed motives of moduli spaces.

Abstract

For proper stacks, unlike schemes, there is a distinction between rational and integral points. Moreover, rational points have extra automorphism groups. We show that these distinctions exactly account for the lower order main terms appearing in precise counts of elliptic curves over function fields, answering a question of Venkatesh in this case. More generally, using the theory of twisted stable maps and the stacky height functions recently introduced by Ellenberg, Zureick-Brown, and the third author, we construct finite type moduli spaces which parametrize rational points of fixed height on a large class of stacks, so-called cyclotomic stacks. The main tool is a correspondence between rational points, twisted maps and weighted linear series. Along the way, we obtain the Northcott property as well as a generalization of Tate's algorithm for cyclotomic stacks, and compute the exact motives of these moduli spaces for weighted projective stacks.