Local and global densities for Weierstrass models of elliptic curves

We prove local results on the p-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equations which are minimal models of semistable elliptic curves over $\mathbb{Q}$ (that is, elliptic curves with square-free conductor) is $1 / \zeta (2) \approx 60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves over $\mathbb{Q}$ is $\zeta (10)/ \zeta (2) \approx 60.85\%$; the density of integral Weierstrass equations which have square-free discriminant is $\Pi_p \left({ 1-\frac{2}{p^2} - \frac{1}{p^3} }\right) \approx 40.89\%$ which is the same (except for a different factor at the prime $2$) as the density of monic integral cubic polynomials with square-free discriminant (and agrees with a 2013 result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over $\mathbb{Q}$ with square-free minimal discriminant is $\zeta (10) \Pi_p \left({ 1-\frac{2}{p^2} - \frac{1}{p^3} }\right) \approx 42.93\%$

The local results derive from a detailed analysis of Tate’s Algorithm, while the global ones are obtained through the use of the Ekedahl Sieve, as developed by Poonen, Stoll, and Bhargava.

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In memory of John Tate, 1925–2019

Received 16 June 2020

Received revised 19 October 2021

Accepted 8 November 2021

Published 13 September 2023