On the density of elliptic curves

We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any alpha > 1/6 (resp. alpha > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture \ Delta (E)\ much less than (alpha) N-E(12 alpha) is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvea and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell-Weil rank 0 and greater than or equal to2, respectively.