Kodaira-Neron statistics for rational elliptic curves with $j$-invariant 0 and 1728

TL;DR

This paper analyzes the distribution of Kodaira-Néron types for elliptic curves over  with specific j-invariants, providing asymptotic counts and related statistics based on height, torsion subgroup, and isogeny-torsion graph.

Contribution

It offers new asymptotic formulas and statistical insights for elliptic curves with j-invariant 0 or 1728, focusing on their reduction types at primes 2 and 3.

Findings

Asymptotic counts for Kodaira-Néron types at primes 2 and 3.

Statistics conditioned on torsion subgroup and isogeny-torsion graph.

Distribution patterns of elliptic curves with special j-invariants.

Abstract

Elliptic curves over $\Q$ with $j$-invariant 0 or 1728 have additive reduction at all primes of bad reduction. In addition, all elliptic curves with $j$-invariant 0 have bad reduction at $p=3$ and all elliptic curves with $j$-invariant 1728 have bad reduction at $p=2$. In this paper we count elliptic curves with $j$-invariant 0 and 1728 by height and determine asymptotics for the various Kodaira-N\'eron types at 3 and 2, respectively. We also give related statistics by holding the torsion subgoup and isogeny-torsion graph constant.