Conductor distributions of elliptic curves

TL;DR

This paper derives an explicit density function for the distribution of conductors of rational elliptic curves relative to their height, providing effective bounds for counting curves with small conductors compared to their height.

Contribution

It offers an explicit density function for conductor distribution of elliptic curves ordered by height, improving bounds on counting curves with small conductors.

Findings

Derived an explicit density function for N/H ratios.

Provided the strongest bounds for counting elliptic curves with small conductors.

Applied results to enumerate curves with conductors much smaller than their height.

Abstract

We determine the distribution of the conductors $N$ of rational elliptic curves when ordered by naive height $H$, in the form of an explicit density function for the ratios $N/H$. Our work is essentially an effective version of the Brumer--McGuinness--Watkins heuristic. Applying our results to the problem of enumerating elliptic curves by conductor gives the strongest bounds yet for the number of elliptic curves which have conductor much smaller than their height for ranges up to $H \ll N^{1.2165}$.