-----Original Message----- From: math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission .com] On Behalf Of James Buddenhagen Sent: Wednesday, February 28, 2007 2:26 PM To: math-fun Subject: [math-fun] irrational eggs
Since eggs are aux currant, let me say that my favorite eggs are the eggs of real elliptic curves. Many years ago I asked Peter Montgomery if the rational points on a real elliptic curve (of positive rank over Q) are dense in the real curve. If I remember correctly, his answer was that they are dense in the connected component containing the identity but that the egg might have no rational points.
So, since this list obviously has readers who know much more than I do, let me ask: suppose y^2 = cubic(x) models an elliptic curve of positive rank over Q, and suppose that the cubic has 3 real roots. How can we tell (say from the coefficients of the cubic) whether or not the real egg has no rational points or is dense with rational points? Specific examples would be really nice.
Why restrict the question of density to the real part of the curve? For a cubic where the rational points are dense in one real component, and missing on the egg, are the rational points dense everywhere in the complex curve? I would guess that the complex rational points are either dense everywhere or nowhere, but I can't see an easy proof of this. Andy Latto andy.latto@pobox.com