ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11G05, 11N36 Wong, Siman Y. Department of Mathematics, Brown University. USA 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 α > 1/6 (resp. α > 1/12) the set of Frey curves (resp. semistable curves) for which the generalized Szpiro conjecture |Δ(E)| ≪_{α} N_{E}^{12α} is false has density zero. 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 Gouvêa-Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell-Weil rank 0 and ≥ 2, respectively.