Elliptic Curves: Part (a) -- Florian Sprung, June 7, 2004

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An elliptic curve over the rationals can be thought of as an equation 
y^2=F(x), where F is a monic polynomial of degree three with rational 
coefficients and distinct complex roots. If we add a point at infinity, 
we can define a geometric group law: namely, we can add two points by 
connecting them via a line, which intersects the curve in a third point. 
It then turns out that we can think of the reflection point of this 
third point about the x-axis as the "sum" of the first two, and the 
point at infinity as the "identity". 

In particular, rational points (points with rational coordinates) form an 
interesting group that is not fully understood today, but has numerous 
applications to number theory. The presenter will talk about two main 
theorems, by Nagell and Lutz, and Mordell and Weil, that describe this group.