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

Let $p\in\{2,3\}$, let $q=p^r$ with $r\geq1$, and put $K=\mathbb F_q(t)$.

We determine the exact number of $K$-isomorphism classes of elliptic curves of bounded Faltings height, equivalently of bounded minimal-discriminant degree.

Two small-characteristic phenomena enter the count.

First, certain generalized Weierstrass equations with nonsmooth generic fiber have a unique geometric singular point defined only after a nontrivial purely inseparable extension of $K$.

Second, the extra $K$-defined automorphisms on the $j=0$ locus, including wild automorphisms, must be incorporated when passing from weighted to unweighted counts.

Building on de Jong's weighted-counting framework, we correct the nonsmooth-locus subtraction, classify normalized fixed pairs with a marked automorphism, and transfer the resulting coefficient-degree counts to exact Faltings height through the intrinsic effective divisor recording minimality defect.

This yields closed unweighted formulas and identifies the geometric origin of every lower-order term.

At height zero, the $j=0$ contribution agrees with the finite-field twist counts of Kronberg-Soomro-Top.