Elliptic Curves

Abstract

These are notes from a first course on elliptic curves at Leiden university in spring 2015. They are aimed at advanced batchelor/beginning master students. We do not assume any backgound in algebraic geometry. We define varieties via functors points, but only on the category of fields. This makes several things simpler, but is not ideal in all respects for example, defining morphisms of varieties as functors doesn’t give what one wants. The main result of the course is a proof of the Mordell-Weil theorem for elliptic curves over Q with rational 2-torsion, via Selmer groups. Our proof of this is fairly complete, except that at one point we have to assume more algebraic geometry to show that non-constant maps of curves are surjective (but this can just be taken as a black box). Not everything from the lectures has been typeset, in particular some examples and basic definitions are omitted. The handwritten notes on the course website are complete, but then you have to read my handwriting! Comments and corrections are very welcome, please email them to David.