In 16-century Fermate stated a conjecture in number theory:
For $(a,b,c,n) \in N\times N\times N \times N$ and $n \geq 3$. The equation $a^n+b^n=c^n$ has no solution.
And claimed he had the proof! Unfortunately, after nearly 3 decades, Wiles was finally able to prove the conjecture. Using sophisticated language(=construction) and techniques from Algebraic Number Theory, Galois Representation, Elliptic Curves, and Complex Analysis.
The story of the proof begins with the “hypothetical” curve named by Gerhard Frey when he found a link between Elliptic Curves and Diophantine Equation as explained in his paper:
Links between stable elliptic curves and certain diophantine equations
He observed a strong connection between Fermate Last Theorem and a conjecture called Taniyama–Shimura conjecture:
All elliptic curves over $\mathbb{Q}$ field are related to modular form.
Roughly speaking, he stated that to counterexample Fermate Last Theorem you can create a hypothesized curve called later Fray Curve, such a curve is not modular. Equivalency and simply put, if the Taniyama–Shimura conjecture is true, the Fermate Last Theorem also is.
Later, Jean-Pierre Serre confirmed this fact and formulated the Epsilon Conjecture which proved by Ken Ribet.
Ribet’s result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat’s Last Theorem - Quoted from Wiles paper.
So, it is enough to show that all semistable elliptic curves over $\mathbb{Q}$ field are modular to prove Fermate Last Theorem.
The proof can be sketched in minor steps, but the technicality required ~110 pages to accomplish. The main breakthrough behind it is:
A new and surprising link between two strong but distinct traditions in number theory, the relationship between Galois representations and modular forms on the one hand and the interpretation of special values of L-functions on the other.
So, how it was accomplished?
First, Wiles started to study Elliptic Curves using their Associated Galois Representation. In simple terms, if you proved that the Associated Galois Representation has a modular form, then the curve is. He picked $p = 3; G(E,3)$ initially, The Galois Representation of the elliptic curve $E$ on $p_3$ division points. Then for some technical reasons, picked $p = 5; G(E,5)$ also.
The second is what he used to generalize his result for all semistable elliptic curves. The Modular Lifting Theorem in which he engaged Class Number Formula to conduct an induction over all elliptic curves, once proved for one curve it automatically extended to all subsequent curves.
Trying to simplify the topic in somehow technical language without being too strict. ↩