Exact counts of elliptic curves of bounded height over $\mathbb F_q(t)$ in characteristics $2$ and $3$

Abstract

Let $p\in\{2,3\}$, let $q=p^r$ with $r\geq1$, and put $K=\mathbb F_q(t)$. We determine the exact weighted and unweighted counts of $K$-isomorphism classes of elliptic curves of bounded Faltings height, equivalently of bounded minimal-discriminant degree. Writing $B$ for the discriminant height bound, the leading term in each count is of order $B^{5/6}$ and has the same coefficient in every characteristic, whereas the lower-order terms depend substantially on $p$ and on the arithmetic of the constant field. In the unweighted count, characteristic two produces a term of order $B^{5/12}$ and, when $r$ is even, a term of order $B^{1/4}$, neither of which occurs for $p>3$. Some characteristic-specific lower-order coefficients are negative. These terms reflect two small-characteristic phenomena. First, the nonsmooth locus of generalized Weierstrass equations contains quasi-elliptic-type strata that are not detected by the rational-singular-section argument in de Jong's count: the unique geometric singular point of the generic fiber may be defined only after a nontrivial purely inseparable extension of $K$. Second, on the $j=0$ locus, the geometric origin-preserving automorphism groups are nonabelian and contain wild elements, while twisting affects which automorphisms descend to $K$. Consequently, the passage from weighted to unweighted counts requires a marked-inertia calculation involving conjugacy classes and centralizer weights. Our formulas show that nonsmooth-locus corrections, extra-automorphism loci, and the removal of minimality defects collectively account for all lower-order terms. Together with the characteristic-greater-than-three formulas of Bejleri-Park-Satriano, this completes the exact weighted and unweighted bounded-height enumerations over $\mathbb F_q(t)$ in every characteristic.