Conductor distributions of elliptic curves

TL;DR

The paper determines the distribution of conductor sizes $N_E$ relative to the naive height $H_E$ for rational elliptic curves, by ordering curves via height and deriving an explicit density for the ratio $N_E/H_E$ in two families $\\mathcal{F}$ and $\\mathcal{F}_6$. The main result expresses the distribution with an archimedean factor $F_\\Delta$ and finite local densities $\\rho(p,n)$ (and $\\rho_6$), yielding a support of $N_E/H_E$ on $(0,496]$ and a continuous, non-differentiable density. It also provides strong bounds for counting elliptic curves by conductor, including a sharp tail bound for $N_E < X$ when $H$ is large, and optimal behavior for $N_E \,\le\, 4464 H/\\log H$. Methodologically, the work makes the Brumer--McGuinness--Watkins heuristic effective through a careful adelic decomposition and a key discriminant-distribution lemma (disc_dist) that handles local restrictions via Möbius inversion and explicit reduction-type data. These results enable translating height-based statistics into conductor statistics and have applications to problems like ratios conjecture.

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 $H \ll N^{1.2165}$.

Figures (2)

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Theorems & Definitions (43)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.6
• Theorem 1.7: Conductor distribution
• Theorem 1.9: Counting by conductor
• Theorem 1.10
• Lemma 2.1
• Lemma 3.1