Local and global densities for Weierstrass models of elliptic curves

Official URL: http://dx.doi.org/10.4310/MRL.2023.v30.n2.a5

We prove local results on the p-adic density of elliptic curves over Qp with different reduction types, together with global results on densities of elliptic curves over 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 Q (that is, elliptic curves with square-free conductor) is 1/ζ(2) ≈ 60.79%, the same as the density of square-free integers; the density of semistable elliptic curves over Q is ζ(10)/ζ(2) ≈ 60.85%; the density of integral Weierstrass equations which have square-free discriminant is Qp (1 - p22 + p13) ≈ 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 2013 result of Baier and Browning for short Weierstrass equations); and the density of elliptic curves over Q with square-free minimal discriminant is ζ(10) Qp (1 - p22 + p13) ≈ 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.