Quadratic twists of elliptic curves

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• John Coates (Cambridge)
• Tuesday 25 February 2014, 16:15-17:15
• MR13.

If you have a question about this talk, please contact James Newton.

In the family of all quadratic twists of an elliptic curve defined over Q, it has long been folklore that one expects that, amongst all twists with root number +1 (resp. root number -1), those whose complex L-function does not vanish at s=1 (resp. whose complex L-function has a simple zero at s=1) should have density 1. This has never been proven for a single elliptic curve over Q, but recently Y. Tian discovered a new method for making important progress in this direction for the quadratic twists of the elliptic curve y² = x³ – x. In my lecture, I shall discuss joint work with Y. Li, Y. Tian and S. Zhuai which makes some first steps towards extending Tian’s method to the quadratic twists of a large family of elliptic curves over Q. Some of our results also generalize an old lemma due to B. Birch.

This talk is part of the Number Theory Seminar series.

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