1
Introduction
2
First reductions of the problem
▶
2.1
Overview
2.2
Reduction to \(n\geq 5\) and prime
2.3
Frey packages
2.4
Galois representations and elliptic curves
2.5
The Frey curve
2.6
Reduction to two big theorems.
3
Elliptic curves, and the Frey Curve
▶
3.1
Overview
3.2
The arithmetic of elliptic curves
3.3
Good reduction
3.4
Multiplicative reduction
3.5
Hardly ramified representations
3.6
The l-torsion in the Frey curve is hardly ramified.
3.7
The l-torsion in the Frey curve is irreducible.
4
An overview of the proof
▶
4.1
Potential modularity.
4.2
A modularity lifting theorem
4.3
Compatible families, and reduction at 3
5
An example of an automorphic form
▶
5.1
Introduction
5.2
A quaternion algebra
5.3
\(\widehat{\mathbb {Z}}\)
5.4
More advanced remarks on \(\widehat{\mathbb {Z}}\) versus \(\mathbb {Q}\)
5.5
\(\widehat{\mathbb {Q}}\) and tensor products.
5.6
A crash course in tensor products
5.7
Additive structure of \(\widehat{\mathbb {Q}}\).
5.8
Multiplicative structure of the units of \(\widehat{\mathbb {Q}}\).
5.9
The Hurwitz quaternions
5.10
Profinite completion of the Hurwitz quaternions
6
Automorphic forms and the Langlands Conjectures
▶
6.1
Definition of an automorphic form
6.2
The finite adeles of the rationals.
6.3
The adelic general linear group
6.4
Smooth functions
6.5
Slowly-increasing functions
6.6
Weights at infinity
6.7
The action of the universal enveloping algebra.
6.8
Automorphic forms
6.9
Hecke operators
7
Miniproject: Frobenius elements
▶
7.1
Introduction and goal
7.2
Statement of the theorem
7.3
The extension \(B/A\).
7.4
The extension \((B/Q)/(A/P)\).
7.5
The extension \(L/K\).
7.6
Proof of surjectivity.
8
Miniproject: Quaternion algebras
▶
8.1
The goal
8.2
Initial definitions
8.3
The adelic viewpoint
8.4
Statement of the main result of the miniproject
8.5
Results about adeles of a number field that we await
8.6
Results about finite adeles which we can work on now
9
Appendix: A collection of results which are needed in the proof.
▶
9.1
Results from class field theory
9.2
Structures on the points of an affine variety.
9.3
Algebraic groups.
9.4
Automorphic forms and representations
9.5
Galois representations
9.6
Algebraic geometry
9.7
Algebra
10
Bibliography
Chapter 1 graph
Chapter 2 graph
Chapter 3 graph
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Chapter 5 graph
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Chapter 7 graph
Chapter 8 graph
Chapter 9 graph
A Blueprint for Fermat’s Last Theorem
Kevin Buzzard, Richard Taylor
1
Introduction
2
First reductions of the problem
2.1
Overview
2.2
Reduction to \(n\geq 5\) and prime
2.3
Frey packages
2.4
Galois representations and elliptic curves
2.5
The Frey curve
2.6
Reduction to two big theorems.
3
Elliptic curves, and the Frey Curve
3.1
Overview
3.2
The arithmetic of elliptic curves
3.3
Good reduction
3.4
Multiplicative reduction
3.5
Hardly ramified representations
3.6
The l-torsion in the Frey curve is hardly ramified.
3.7
The l-torsion in the Frey curve is irreducible.
4
An overview of the proof
4.1
Potential modularity.
4.2
A modularity lifting theorem
4.3
Compatible families, and reduction at 3
5
An example of an automorphic form
5.1
Introduction
5.2
A quaternion algebra
5.3
\(\widehat{\mathbb {Z}}\)
5.4
More advanced remarks on \(\widehat{\mathbb {Z}}\) versus \(\mathbb {Q}\)
5.5
\(\widehat{\mathbb {Q}}\) and tensor products.
5.6
A crash course in tensor products
5.7
Additive structure of \(\widehat{\mathbb {Q}}\).
5.8
Multiplicative structure of the units of \(\widehat{\mathbb {Q}}\).
5.9
The Hurwitz quaternions
5.10
Profinite completion of the Hurwitz quaternions
6
Automorphic forms and the Langlands Conjectures
6.1
Definition of an automorphic form
6.2
The finite adeles of the rationals.
6.3
The adelic general linear group
6.4
Smooth functions
6.5
Slowly-increasing functions
6.6
Weights at infinity
6.7
The action of the universal enveloping algebra.
6.8
Automorphic forms
6.9
Hecke operators
7
Miniproject: Frobenius elements
7.1
Introduction and goal
7.2
Statement of the theorem
7.3
The extension \(B/A\).
7.4
The extension \((B/Q)/(A/P)\).
7.5
The extension \(L/K\).
7.6
Proof of surjectivity.
8
Miniproject: Quaternion algebras
8.1
The goal
8.2
Initial definitions
8.3
The adelic viewpoint
8.4
Statement of the main result of the miniproject
8.5
Results about adeles of a number field that we await
8.6
Results about finite adeles which we can work on now
9
Appendix: A collection of results which are needed in the proof.
9.1
Results from class field theory
9.2
Structures on the points of an affine variety.
9.3
Algebraic groups.
9.4
Automorphic forms and representations
9.5
Galois representations
9.6
Algebraic geometry
9.7
Algebra
10
Bibliography