Local and global densities for Weierstrass models of elliptic curves

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 $1/\zeta(2)\approx60.79\%$, the same as the density of square-free integers; the density of semistable elliptic curves over $\mathbb{Q}$ is $\zeta(10)/\zeta(2)\approx60.85\%$; the density of integral Weierstrass equations which have square-free discriminant is $\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right) \approx 42.89\%$, 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 previous result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over $\mathbb{Q}$ with square-free minimal discriminant is $\zeta(10)\prod_p\left(1-\frac{2}{p^2}+\frac{1}{p^3}\right)\approx42.93\%$. 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.