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

Jun-Yong Park

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 \(\text{char}(k) \ge 0\).

Keywords: elliptic curves, families over a function field, counting.

AMS Subject Classification: Primary 14H10, 14H52, 14G17.