Learning Elliptic Curves

The simple form of an elliptic curve (Weierstrass equation) From Wikipedia
Two elliptic curves. From Wikipedia

‘Algebraic Number Theory’ can be read in two distinct ways. One
is the theory of numbers viewed algebraically, the other is the study of
algebraic numbers. Both apply here. We illustrate how basic notions from
the theory of algebraic numbers may be used to solve problems in number
theory. However, our main focus is to extend properties of the natural
numbers to more general number structures: algebraic number fields, and
their rings of algebraic integers. These structures have most of the standard
properties that we associate with ordinary whole numbers, but some subtle
properties concerning primes and factorization sometimes fail to generalize.

Part IV describes the final breakthrough, when — after a long period
of solitary thinking — Wiles finally put together his proof of Fermat’s Last
Theorem. Even this tale is not without incident. His first announcement
in a lecture series in Cambridge turned out to contain a subtle unproved
assumption, and it took another year to rectify the error. However, the
proof is finally in a form that has been widely accepted by the mathematical
community. In this text we cannot give the full proof in all its glory.
Instead we discuss the new ingredients that make the proof possible: the
ideas of elliptic curves and elliptic integrals, and the link that shows that
the existence of a counterexample to Fermat’s Last Theorem would lead
to a mathematical construction involving elliptic integrals. The proof of
the theorem rests upon showing that such a construction cannot exist. We
end with a brief survey of later developments, new conjectures, and open
problems.

In this chapter we introduce the important notion of an ‘elliptic curve’.
Elliptic curves are a natural class of plane curves that generalise the straight
lines and conic sections studied in nearly all university mathematics courses
(and many high school courses). However, the study of elliptic curves
involves two new ingredients. First, it is useful to consider complex curves,
not just real ones. Second, for some purposes it is more satisfactory to work
in complex projective space rather than the complex ‘plane’ C2. (Algebraic
geometers call C the complex line because it is l-dimensional over C.) We
shall introduce these refinements in simple stages.

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