As stated in a much earlier post, one of my pet projects is learning about Fermat's last theorem.
Understanding the full proof is honestly beyond my ability but what I would like to do in this sequence of posts is to explain some of the concepts that are involved in the proof.
Essentially, there are three key concepts:
1) Elliptic curve: This is known as a variety in the beautiful branch of mathematics called algebraic geometry. Essentially, an elliptic curve is the solution set of a polynomial equation. Concretely, in the simplest case, it can be described via a Weierstrass equation of the form:
![\[ y^2 = x^3 + ax + b. \]](https://www.peterjong.com/wp-content/ql-cache/quicklatex.com-d375e49c5a074b14f6d1a1334baafddf_l3.png "Rendered by QuickLaTeX.com")
For a given pair of complex numbers 
, the set of ordered pairs 
of complex numbers satisfying the given equation takes on the shape of a torus (or a donut in plain language). The points on the curve form a group under a well-known operation.
2) Modular forms: These are complex-valued functions satisfying many symmetries. For example, modular functions, which may considered as modular forms of weight 0, satisfy
![\[ f\bigg(\frac{az+b}{cz+d}\bigg)=f(z), \]](https://www.peterjong.com/wp-content/ql-cache/quicklatex.com-29c7b9aee37c9f194067e30f19a11c2d_l3.png "Rendered by QuickLaTeX.com")
where
![\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}_2(\mathbf{Z}). \]](https://www.peterjong.com/wp-content/ql-cache/quicklatex.com-c6dab09efaa44f39d1a3d97b14445919_l3.png "Rendered by QuickLaTeX.com")
Thus, modular functions are invariant under area-preserving transformations such as rotations or reflections.
3) Galois representations: This represents the link between modular forms and elliptic curves. Essentially, a Galois representation is a homomorphism of a Galois group such as 
to an automorphism group of a vector space.
An example of a Galois representation of relevance to us is the following. For an elliptic curve defined as in 1) above, with 
and 
rationals, the 
-torsion points form a subgroup which turns out in general to be isomorphic to 
. The Galois group permutes these torsion points resulting in a representation:
![\[\rho_E: \text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_2(\mathbb{Z}/n\mathbb{Z}), \]](https://www.peterjong.com/wp-content/ql-cache/quicklatex.com-26805781811a35a881688abe0cb6d1a4_l3.png "Rendered by QuickLaTeX.com")
on identifying a basis of the -torsion points.
Comments are closed