ETD

• arithmetic geometry
• elliptic curves
• Fermat's Last Theorem
• modular forms
• modularity
• number theory

This master's thesis explores some aspects of the proof of Fermat's Last Theorem, a problem that challenged mathematicians for over 350 years. The proof, famously completed by Andrew Wiles with the assistance of Richard Taylor, is a major achievement in modern mathematics and opened new research directions, and even today, after 30 years, its methods continue to have an impact.
The story of the proof begins in the 1980s with an insight from Gerhard Frey. He proposed that if a non-trivial solution to Fermat's equation existed, it would be possible to construct a semistable elliptic curve—now known as the Frey curve—that would be a counterexample to the Taniyama-Shimura Conjecture (now a theorem), which states that every elliptic curve defined over the rational numbers is modular. Building upon a conjecture by Jean-Pierre Serre, it was Ken Ribet who confirmed Frey's suspicion and proved that this hypothetical Frey curve could not be modular. This result, known as Ribet's Theorem, definitively linked the truth of Fermat's Last Theorem to the Modularity Theorem.
The central challenge, then, was to prove the Modularity Theorem, at least for the class of semistable elliptic curves. Andrew Wiles undertook this in \cite{Wiles}. His strategy transformed this problem from one of geometry and number theory into one of commutative algebra. Wiles constructed two distinct but related mathematical objects: a universal deformation ring $R$ and a Hecke ring $T$. The ring $R$ parametrizes families of Galois representations, including one associated with $E$. On the other hand, $T$ is constructed from modular forms and their associated Hecke operators and parametrizes families of Galois representations that are modular.
Wiles' contribution was to prove that these rings are isomorphic, a result now famously known as the $R=T$ theorem. This isomorphism establishes that any Galois representation arising from a semistable elliptic curve is, in fact, modular. This result completed the proof of the Modularity Theorem for semistable elliptic curves, and by extension, the proof of Fermat's Last Theorem.
This thesis aims to present the construction of the two rings $R$ and $T$ and to provide an exposition of some of the key mathematical machinery required to understand Wiles's proof. The work is divided into five chapters:
In Chapter~$1$, we lay the groundwork by introducing the theory of $\ell$-adic Galois representations, focusing on the one arising from the Tate module of abelian varieties.
Chapter~$2$ then delves into Mazur's deformation theory (following \cite{MazurDefCornell}), which provides the framework for constructing the universal deformation ring $R$. This ring is defined as a quotient of a ring representing a functor: Mazur's deformation functor. Following the exposition in \cite{Gouvea}, we show that this functor satisfies the hypotheses of a theorem by Schlessinger \cite{Schlessinger}, which ensures that the functor is indeed representable.
Chapter~$3$ covers the essential theory of modular forms and Hecke operators. A significant portion is dedicated to interpreting Hecke operators as algebraic correspondences, and the consequent induced action of Hecke operators on the homology and cohomology of modular curves. In particular, we prove that the algebra of Hecke operators is finitely generated. Following \cite{SteinModFormsCompapproach}, we also discuss the theory of modular symbols, which is behind the implementation of modern algorithms to compute newforms.
In Chapter~$4$, using the tools introduced in the previous chapter, we construct the Hecke ring $T$. While the ring $R$ is defined by means of a universal property, $T$ has a much more concrete construction and can be explicitly computed. Following \cite{LarioSchoof}, we provide an example of such a computation.
Finally, Chapter~$5$ culminates in an exposition of the commutative algebra criterion for isomorphism of complete intersection, as presented in \cite{LenstraCI}, used to prove the isomorphism between $R$ and $T$.