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On the Density of Elliptic Curves

Published online by Cambridge University Press:  04 December 2007

Siman Wong
Siman Wong
Affiliation:
Department of Mathematics, Brown University. Providence, RI 02912, U.S.A. Current address: Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, U.S.A. E-mail: siman@math.umass.edu
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Abstract

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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. all elliptic curves) for which the generalized Szpiro Conjecture |Δ(E)| [Lt]α NE12α 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 Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and [ges ]2, respectively.

Keywords

elliptic curvesheightquadratic twistsranksroot numberssquare-free sieve
Type
Research Article
Information
Compositio Mathematica , Volume 127 , Issue 1 , May 2001 , pp. 23 - 54
Copyright
© 2001 Kluwer Academic Publishers