Counting isomorphism classes of elliptic curves over $\mathbb{F}_q(t)$

TL;DR

This paper precisely counts the isomorphism classes of elliptic curves over rational function fields over finite fields of characteristic 2 and 3, using advanced stack and motivic techniques.

Contribution

It provides an exact enumeration of elliptic curve classes over $_q(t)$ by computing rational points on specific classifying stacks, extending classical results.

Findings

Exact counts of elliptic curve isomorphism classes over $_q(t)$ for char 2,3.

Methodology based on rational points on classifying stacks and motivic height zeta functions.

Complete classification over any rational function field with perfect residue field.

Abstract

We determine the precise number of isomorphism classes of elliptic curves over $\mathbb{F}_q(t)$ with $\text{char}(\mathbb{F}_q) = 3,2$. The key idea is to obtain the exact unweighted number of rational points on the classifying stacks $\mathcal{B} Q_{12}$, $\mathcal{B} Q_{24}$ and $\mathcal{B} Z$, where $Q_{12}$ and $Q_{24}$ denote the dicyclic groups of orders 12 and 24, respectively, and $Z$ denotes the non-reduced group scheme of order 2. This computation, inspired by the classical work of [de Jong] and performed via motivic height zeta functions of height moduli spaces constructed in [Bejleri-Park-Satriano], establishes a complete determination of the total number of isomorphism classes of rational points on $\overline{\mathcal{M}}_{1,1}$ over any rational function field $k(t)$ with perfect residue field $\text{char}(k) \ge 0$.