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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 10

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ELLIPTIC CURVES OVER Q

Road Map

We have defined a particularly interesting finitely generated abelian group structure on the points of an elliptic curve. We now can study the torsion and rank of this group. In this way, we are beginning to close in on our quarry, counting the rational solutions to a cubic equation, or equivalently counting the rational points on an elliptic curve over Q.

1. The Basic Structure of the Group

In chapter 8, we took an equation of the form y2 = x3 + Ax + B, and discussed how to turn the set of solutions to this equation, along with , into a group. In that chapter, our examples mostly used coefficients ...

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