The Proof of Fermat’s Last Theorem

Abstract . The goal of this post is to sketch the proof of Fermat’s Last Theorem. We remind readers that this post is currently incomplete and is still being updated.

Contents
Contents
 1.  Introduction
 2.  The proof of the Shimura-Taniyama-Weil conjecture
 3.  The proof of the Modularity Lifting Theorem
   3.1.  The main theorem
   3.2.  Wile’s numberical criterion

1. Introduction

Theorem 1 (Fermat’s Last Theorem=$(FLT)( n) $). For $ n>2 $, the fermat’s equation $ x^{n}+y^{n}=z^{n} $ has no non-zero integer solutions.

The cases of $ n=3,4 $ were proved by Euler and Fermat respectively. We refer to the link https://zhuanlan.zhihu.com/p/39616274 for a complete proof. Of course $ (FLT)( n_1) $ implies $ (FLT)( n_2) $ if $ n_1 |n_2$, and so it suffices to prove the following

Theorem 2. If $\ell\geq 5 $ is a prime, then $ a^{\ell}+b^{\ell}=c^{\ell} $ has no nonzero integer solutions.
Proof .

  1. (Frey curve) Suppose that there’s a solution $ (a,b,c) $ to the Fermat’s equation. Let us consider the Frey curve $ E=E_{a^{\ell},b^{\ell},c^{\ell}}: y^{2}=x(x-a^{\ell})(x+b^{\ell}) $,it’s a elliptic curve over $ \mathbb{Q} $ satisfying (1) it’s semistable with conductor $ N=\prod_{p|abc}p$ (where $ p$ runs over primes), (2) one can construct a Galois representation $ \bar{\rho}_{E,\ell}:G_{\mathbb{Q}}=Gal(\bar{\mathbb{Q}}/\mathbb{Q})\to \text{Aut}(E(\ell))\simeq GL_{2}(\mathbb{F}_{\ell})$.

  2. (Shimura-Taniyama-Weil conjecture: Every elliptic curve over $ \mathbb{Q}$ is modular.) For any elliptic curve $ E$ over rational number field, there exists a weight two cuspidal modular eigenform $ f_{E}(q)=\sum_{n\geq 1}a_{n}q^{n}$ such that $ a_{p}=p+1-|E(\mathbb{F}{p})|$ for any prime $ p$. Now, appying Shimura-Taniyama-Weil conjecture to the Frey curve $ E$ in Step 1, we see that there is a modular form $ f$ such that $ \bar{\rho}_{f}=\bar{\rho}_{E,\ell}$, where $ \bar{\rho}_{f}:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\to GL_{2}(\mathbb{F}_{\ell})$ is the associated Galois representation corresponding to $ f$.

  3. (Serre + Ribet’s work: lowering the level) Shimura-Taniyama-Weil conjecture predicts that the representation $ \bar{\rho}_{E,\ell}$ arises from a modular form of weight two, albeit a form whose level is quite large. (it’s the product of all the primes dividing $ abc $ by Step 1). Ribet proved, if this were the case, then $ \bar{\rho}_{E,\ell}$ would also be associated with a modularform mod $ \ell$ of weight $ 2$ and level $ 2$, in the way predicted by Serre’s conjecture. That is, there is a weight two new form $ g$ of conductor $ 2$ such that $ \bar{\rho}_{g}=\bar{\rho}_{E,\ell}$. But the dimension of $ S_{2}(\Gamma_{0}(2)) $ is eual to the genus of $ X_{0}(2) $, which is easily to be zero. This is a contradiction and Fermat’s Last Theorem is proved.

Warning: In the rest of this post our goal is to prove the Shimura-Taniyama-Weil conjecture.

2. The proof of the Shimura-Taniyama-Weil conjecture

Recall that there are a number of equivalent ways of defining modularity of elliptic curves. Here are a few.

Theorem 3. Let $ E$ be an elliptic curve over $ \mathbb{Q}$, then the following are equivalent to $ E$.

  • There is a weight two newform $ f $of condunctor $ N_{E} $ and trivial character for which $ L(f,s)=L(E,s) $.
  • For some prime $ \ell $, $ \rho_{E,\ell} $ is modular.
  • For every prime $ \ell $, $\rho_{E,\ell} $ is modular.
  • There is a non-constant morphism $ \pi:X_{0}(N_{E})\to E $ of algebraic curves defined over $ \mathbb{Q} $

First we state Wile’s celebrated modularity lifting theorem, which lies at the heart his strategy:

Theorem 4 (Modularity Lifting Theorem). Let $ \rho:G_{\mathbb{Q}}\to GL_{2}(\mathbb{Z}_{\ell}) $ be an irreducible geometric Galois representation satisfying a few technical conditions. If $ \bar{\rho} $ is modular and irreducible, then so is $ \rho $.

Let us first explain how Wiles himself parlays his orginal (semistable) modularity lifting theorem into a proof of the Shimura-Taniayama-Weil conjecture for semistable elliptic curves.

Given such an elliptic $ E $. One obtains the associated Galois representations
$ \bar{\rho}_{E,3}:G_{\mathbb{Q}}\to GL_{2}(\mathbb{F}_{3}) $ and $ \rho_{E,3}:G_{\mathbb{Q}}\to GL_{2}(\mathbb{Z}_{3}) $. The theorem of Langlands and Tunnel about the modularity of the general quartic equation leads to the conclusion that $ \bar{\rho}_{E,3}$ is modular.

If $ E $ is semistable, Wiles is able to check that both $ \rho_{E,3}$ and $ \bar{\rho}_{E,3}$ satisfy the conditions necessary to apply the Modularity Lifting Theorem, at least when $ \bar{\rho}_{E,3} $ is irreducible. It then follows that $ \rho_{E,3}$ is modular.

Having established the modularity of all semistable elliptic curves $ E $ for $ \rho_{E,3}$ is irreducible, Wiles disposes of the others by applying his lifting theorem to prime $ \ell=5 $ instead of $ \ell=3 $. The Galois representation $ \rho_{E,5}$ is always irreducible in this setting, because no elliptic curve over $ \mathbb{Q} $ can have a rational subgroup of order $ 15 $. Nonetheless, the approach of exploiting $ \ell=5 $ seems hopeless at first glance, because the Galois representation $ E[5] $ is not known to be modular a priori.

To establish the modularity of $ E[5] $, Wiles constructs an auxiliary semistable elliptic curve $ E’ $ satisying\[ \rho_{E’,3}=\rho_{E,5} \]where $ \bar{\rho}_{E’,3} $ is irreducible. It then follows from the argument in the previous paragraph that $ E’ $ is modular, hence $ E'[5]=E[5] $ is modular as well, putting $ E $ within striking range of the modularity lifting theorem $ \ell=5 $. This lovely epilogue of Wile’s proof, which came to be known as the “3-5 switch,” may have been viewed as an expedient trick at the time.

Warning: In the rest of this post our goal is to prove the Modularity Lifting Theorem.

3. The proof of the Modularity Lifting Theorem

By a two dimensional galois representation over a topological ring $ A $ we mean a continuous group homomorphism $ \rho:G_{\mathbb{Q}}\to GL_{2}(A) $. We may assume the topological $ A $ is a complete noetherian local ring with finite residue field of characteristic $ p $ (which is what Mazur calls a coefficient ring).

Let $ A $ be a conefficient ring with maximal ideal $ \mathfrak{m}_{A} $ and let $ k_{A} $ be the residual field. We define the residual representation of a a galois representation $ \rho:G_{\mathbb{Q}}\to GL_{2}(A) $ to the representation
\[ \bar{\rho}:G_{\mathbb{Q}}\to GL_{2}(k_{A}) \]
obtained by composing $ \rho $ with the reduciton map $ GL_{2}(A)\to GL_{2}(k_{A}) $. Conversely, if $ \rho_{0}:G_{\mathbb{Q}} \to GL_{2}(k)$ is a two dimensional galois representation over a finite field $ k $, then we say that $ \rho $ is a lifting to $ \rho_{0} $ to $ A $ if $ k=k_{A} $ and $ \bar{\rho}=\rho_{0} $. Two lifting $ \rho,\rho’ $ of $ \rho_{0} $ to $ A $ are said to be equivalent if $ \rho’ $ can be conjugated to $ \rho $ by a matrix in $ GL_{2}(A) $ that is congruent to the identity matrix modulo $ \mathfrak{m}_{A} $??

A deformation of $ \rho_{0} $ to $ A $ is an equivalence class of lifting of $ \rho_{0} $ to $ A$. For a given lifting $ \rho $ of $ \rho_{0} $, we will abuse notation and also write $ \rho $ to denote the deformation to which it belongs.

Let $ k $ be a finite field of characteristic $ p $ and let $ \rho_{0}:G_{\mathbb{Q}}\to GL_{2}(k) $ be a galois representiaon. A deformation type $ \mathcal{D} $ is a list of conditions to be imposed on deformations of a residual representaion $ \rho_{0}:G_{\mathbb{Q}}\to GL_{2}(k) $. We say that a deformation $ \rho $ of $ \rho_{0} $ is of type $ \mathcal{D} $ if the following conditions are satisfied???

Using Mazur’s theory of deformations of galois representations ,Wiles assocaites to each deformation type $ \mathcal{D} $ a universal deformation ring $ R_{\mathcal{D}}$ and a universal defromation
\[ \rho_{D}:G_{\mathbb{Q}}\to GL_{2}(R_{\mathcal{D}}) \]
of $ \rho_{0} $ of type $ D $. The representation $ \rho_{\mathcal{D}} $ satisfies the following universal property: for every deformation $\rho:G_{\mathbb{Q}}\to GL_{2}(A) $ of $ \rho_{0} $ of type $ \mathcal{D} $ there is a unique homomorphism $ \pi_{A}:R_{\mathcal{D}}\to A $ such that the diagrm
???
is commuative.

On the other hand, Wiles defines another coefficient ring $ \mathbf{T}_{\mathcal{D}} $, the universal modular deformation ring and a universal modular deformation
\[ \rho_{\mathcal{D},mod}:G_{\mathbb{Q}}\to GL_{2}(\mathbf{T}_{\mathcal{D}}) \]
of $ \rho_{0} $ of type $ \mathcal{D} $. The representaiton $ \rho_{\mathcal{D},mod} $ satisfies the analogous universal property for modular deformations of type $ \mathcal{D} $.

3.1. The main theorem By the universal property of $ \rho_{\mathcal{D}} $ there is unique homorphism $ \varphi_{\mathcal{D}}:R_{\mathcal{D}}\to \mathbf{T}_{\mathcal{D}} $ such that $ \rho_{\mathcal{D},mod}=\varphi_{\mathcal{D}}\circ \rho_{\mathcal{D}} $. The following theorem is a special case of the main theorem of Wiles.

Theorem 5. Suppose $ \rho_{0} $ satisfies some hypotheses?. Then the canonical map $ \varphi_{\mathcal{D}}:R_{\mathcal{D}} \to \mathbf{T}_{\mathcal{D}}$ is an isomorphism of complete intersection rings.

Corollary 6. Suppose $ \rho_{0} $ satisfies some hypotheses?. Then every deformation of $ \rho_{0} $ of type $ \mathcal{D} $ is modular.

3.2. Wile’s numberical criterion Let $ R $ and $ T $ be coefficient rings and suppose we have a commuative diagram??

is which $ \mathcal{O} $ is a complete discrete valuation ring and all the arrows are surjective. Let $ I_{R}:=\textbf{ker}\pi_{R} $, $ I_{T}:=\textbf{ker}\pi_{T} $, and let $ \eta_{T}:=\pi_{T}(\textbf{Ann}_{T}(I_{T}))$. Then the following three assertions are equivalent.

  • $ \varphi $ is an isomorphism of complete intersection rings.
  • $ I_{R}/I^{2}_{R} $ is finite and $ |I_{R}/I^{2}_{R}\leq |\mathcal{O}/\eta_{T}| $;
  • $ I_{R}/I^{2}_{R} $ is finite and ???

To prove $ \varphi $ is an isomorphism, Wiles establishes the middle inequality in the above criterion. For this, he first interprets the two sides of the inequality in terms of other objects that have been studied in some detail in the literature. More precisely, Wiles interprets the “tangent space” $ \textbf{Hom}_{\mathcal{O}}(I_{R_{\mathcal{D}}}/I^{2}_{R},K/\mathcal{O}) $ as a Selmer group $ ??? $, i.e. as a certain subgroup of the galois cohomology group ???

The proof of the crucial numberical inequality divides into two parts. The minimal case, is proved by Wiles with with Taylor. The non-minimal case is proved by induction on the ?

TBC

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