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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 7

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ABELIAN GROUPS

Road Map

One of the most crucial things about an elliptic curve E defined by an equation with integer coefficients is that E(F) is an abelian group for every field F over which the equation is nonsingular. So we have to explain what an abelian group is before we begin to dig deeper into elliptic curves. In this chapter, we explore the concept of abelian group, and define a key number associated to many abelian groups: the rank, which measures “how big” the group is.

1. How Big Is Infinity?

When we count mathematical objects such as the number of solutions to an equation, the first fundamental distinction we might make ...

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