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

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.