Counting elliptic curves with bad reduction over a prescribed set of primes

TL;DR

This paper estimates the distribution of elliptic curves over rationals with specific bad reduction types at primes and analyzes their conductors, providing explicit proportions and insights into their arithmetic properties.

Contribution

It introduces a method to estimate the proportion of elliptic curves with given Kodaira types at primes and analyzes their conductor properties, advancing understanding of their distribution.

Findings

Approximately 60% of elliptic curves have squarefree conductors outside primes 2 and 3.

The proportion of curves with multiplicative reduction at p is given by a specific rational function.

The distribution of Kodaira types at primes is explicitly quantified for elliptic curves ordered by height.

Abstract

Let $p\ge5$ be a prime and $T$ a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over $\mathbb Q$ up to height $X$ with Kodaira type $T$ at $p$. This enables us find the proportion of elliptic curves over $\mathbb Q$, when ordered by height, with Kodaira type $T$ at a prime $p\ge5$ inside the set of all elliptic curves. This proportion is a rational function in $p$. For instance, we show that $\displaystyle\frac{p^8(p-1)}{p^9-1}$ of all elliptic curves with bad reduction at $p$ are of multiplicative reduction. Furthermore, we prove that the prime-to-$6$ part of the conductors of a majority ($=\zeta(10)/\zeta(2)\approx 0.6$) of elliptic curves are squarefree, where $\zeta$ is the Riemann-zeta function.