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# Mathematical Research Letters

## Volume 30 (2023)

### Number 2

### Local and global densities for Weierstrass models of elliptic curves

Pages: 413 – 461

DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n2.a5

#### Authors

#### Abstract

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.

In memory of John Tate, 1925–2019

Received 16 June 2020

Received revised 19 October 2021

Accepted 8 November 2021

Published 13 September 2023