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

2025/07/09 by Jun-Yong Park, Park, Jun-Yong
#math.NT #math.AG #math.KT

paper · pdf · doi:10.48550/arxiv.2507.06754

Abstract

Let p∈\2,3\, let q=p^r with r≥1, 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.

Related

• On the ranks of elliptic curves with isogenies
• Counting elliptic curves of bounded Faltings height
• Faltings elliptic curves in twisted \mathbb Q-isogeny classes
• Counting elliptic curves with an isogeny of degree three
• Counting 5-isogenies of elliptic curves over ℚ