Proof of Fermat Last Theorem FROM SCRATCH
Summary
This video provides a comprehensive explanation of Fermat's Last Theorem's proof, starting from high school math and progressively building the necessary tools from linear algebra, abstract algebra, and number theory. It delves into elliptic curves, modular forms, Galois representations, and their connections, culminating in the proof of Fermat's Last Theorem using the modularity theorem and related concepts like the Frey curve and Richard's theorem.
Key Insights
The proof relies on connecting elliptic curves and modular forms.
The core idea behind the proof is the connection between two seemingly unrelated mathematical objects: elliptic curves and modular forms. The explanation will build the mathematical tools needed step-by-step.
Introduces the absolute Galois group of Q.
Gal(Q̄/Q) is the group of automorphisms of the algebraic closure of Q that fix Q. It's a vast, complex group central to modern number theory, studied via its actions on modules (Galois representations).
Explains Galois representations translating Galois group actions to module automorphisms.
A Galois representation maps elements of the absolute Galois group (e.g., Gal(Q̄/Q)) to automorphisms of a module (e.g., Tate module of an elliptic curve). This makes the abstract Galois group's actions more concrete via matrices.
Elliptic curve points form an Abelian group with geometric addition.
Points on an elliptic curve form an Abelian group under a geometric addition law: draw a line through P and Q, find the third intersection R, reflect R through the x-axis (for standard forms) to get P+Q. The point at infinity acts as the identity.
Frey curve construction links Fermat's equation to elliptic curves.
Assuming a non-trivial solution aᵖ + bᵖ + cᵖ = 0 leads to the Frey family of elliptic curves y² = x(x-aᵖ)(x+cᵖ). This construction is crucial for the proof.
Frey curves are semi-stable, implying modularity.
The Frey curve is proven to be semi-stable (good or multiplicative reduction at all primes). Its semi-stability guarantees, via the Modularity Theorem, that it corresponds to a modular form.
Hecke forms are simultaneous eigenvectors of all Hecke operators.
A Hecke form is a modular form (often normalized) that is an eigenvector for every Hecke operator T<0xE2><0x82><0x99>. Its coefficients are multiplicative.
Matching L-functions is a key clue suggesting a connection.
The L-functions derived from point counts of elliptic curves (over finite fields) and from the coefficients of Hecke forms exhibit identical structures and formulas, particularly for weight 2 modular forms.
Galois representations provide an algebraic link.
Both elliptic curves (via Tate module) and modular forms (via Hecke forms) can be used to construct Galois representations. The equivalence of these representations implies a deep connection.
The Modularity Theorem states the equivalence.
Every elliptic curve over Q is modular, meaning it corresponds to a weight 2 newform whose level matches the elliptic curve's conductor, and whose L-function/Galois representation matches that of the elliptic curve.
Frey curve's properties link to modularity.
The Frey curve, derived from a hypothetical Fermat solution, is semi-stable, guaranteeing its modularity. This connection is central to Wiles's proof strategy.
Ribet's Theorem connects Galois representations and modular forms.
Ribet's theorem implies that an irreducible mod p Galois representation arising from an elliptic curve must come from a modular form of the same level (conductor).
Construct the Frey curve from the hypothetical solution.
The Frey curve y² = x(x-aᵖ)(x+cᵖ) is constructed based on the assumed solution. Properties of a, b, c ensure specific characteristics of this curve.
The Frey curve is semi-stable.
Analysis of the Frey curve's discriminant and related quantities shows it has good or multiplicative reduction at all primes, meaning it is semi-stable.
Semi-stability implies modularity by the Modularity Theorem.
Since the Frey curve is semi-stable, the Modularity Theorem guarantees it corresponds to a weight 2 newform of level equal to the curve's conductor.
Ribet's Theorem connects the Frey curve's Galois representation to modular forms.
Ribet's Theorem states that if a mod p Galois representation associated with the Frey curve is modular (which it is due to semi-stability), it must come from a modular form of level 2.
Contradiction arises from the level of the modular form.
The Frey curve's conductor determines the level of the corresponding modular form. Analysis shows this conductor must be 2. However, the structure of the Frey curve implies its corresponding modular form should have a level related to 'abc', which cannot be 2.
The contradiction proves the initial assumption is false.
The existence of a modular form at level 2 contradicts the properties derived from the hypothetical Fermat solution (which implies a higher level). Therefore, the initial assumption of a non-trivial solution must be false.
Sections
Introduction
Fermat's Last Theorem is legendary for its proof, not its story.
The video aims to bridge the gap between overly simplified explanations of Fermat's Last Theorem and overly complex mathematical treatments. It promises to explain the proof starting from high school math, introducing concepts incrementally.
The proof relies on connecting elliptic curves and modular forms.
The core idea behind the proof is the connection between two seemingly unrelated mathematical objects: elliptic curves and modular forms. The explanation will build the mathematical tools needed step-by-step.
The video uses a top-down approach where each slide serves a purpose.
Every slide is purposefully included, foreshadowing concepts needed later in the proof. For instance, partial derivatives are needed for singular points on elliptic curves, which connects to reduction types and L-functions.
The video is structured into six chapters.
Chapter 1: Basics (modulo functions, notation, n=3 and n=4 cases of FLT). Chapter 2: Linear Algebra. Chapter 3: Abstract Algebra (groups, rings, fields, Galois theory). Chapter 4: Elliptic Curves. Chapter 5: Modular Forms. Chapter 6: The Endgame (connecting everything for the proof).
Chapter 1: Basics
Familiarizes viewers with common number sets and notation.
Introduces standard notations for natural numbers (N), integers (Z), rational numbers (Q), real numbers (R), and complex numbers (C), emphasizing set inclusion relationships. It also explains Cartesian products like R^2.
Explains essential mathematical symbols and abbreviations.
Covers symbols like 'for all' (∀), 'there exists' (∃), 'defined as' (:=), 'without loss of generality' (WLOG), 'such that' (s.t.), 'want to show' (WTS), and 'enough to show' (ETS).
Fermat's Last Theorem stated using mathematical notation.
Fermat's Last Theorem ( fyrir n ≥ 3, there does not exist xyz ∈ Z such that xⁿ + yⁿ = zⁿ, xyz ≠ 0) is presented using the introduced symbols.
Defines divisibility and congruence notation.
Explains 'a divides b' (a|b) meaning b = ak for some integer k, 'a does not divide b' (a<0xC2><0x84>b), and congruence modulo n (a ≡ b mod n) meaning n divides (a-b), equivalent to having the same remainder when divided by n.
Explains properties of greatest common divisors (GCD).
Covers notation for GCD (gcd(a,b) or (a,b)) and introduces three useful properties: gcd(a,c) = gcd(b,c) if gcd(a,c)=1; gcd(a,b) = gcd(a, b-ak); and if gcd(a,b)=1 and a|bc, then a|c. These properties are explained intuitively.
Proves Fermat's Last Theorem for n=3 using a lemma.
A lemma (which will be proven later in abstract algebra) states: If p, q are coprime integers and p² + 3q² = s³, then there exist coprime integers u, v such that u³ + 3v³ = 4q³ and p = u² - 3v². Using this lemma and infinite descent, the n=3 case is proven by assuming a solution and constructing a smaller solution, leading to a contradiction.
Proves Fermat's Last Theorem for n=4 using a lemma.
A lemma states: No right triangle with integer sides can have an area that is a perfect square. The proof uses infinite descent, assuming a solution and constructing a smaller one, leading to a contradiction. It involves analyzing squares modulo 4 and properties of Pythagorean triples.
Chapter 2: Linear Algebra
Introduces matrices as rectangular arrays of numbers.
Defines matrices by their dimensions (m x n) and how to denote entries (Aij). Explains matrices over fields like R (real numbers) and C (complex numbers).
Explains basic matrix operations: addition, subtraction, and scalar multiplication.
Matrix addition/subtraction is entry-wise for matrices of the same size. Scalar multiplication involves multiplying every entry by the scalar.
Details matrix multiplication, emphasizing its non-commutativity.
Matrix multiplication (AB) requires the number of columns in A to equal the number of rows in B. The resulting matrix has dimensions (rows of A) x (columns of B). The entry (AB)ij is the dot product of row i of A and column j of B. AB is generally not equal to BA.
Defines special types of matrices.
Covers square matrices (rows=cols), zero matrices (additive identity), diagonal matrices (non-diagonal entries are zero), and identity matrices (diagonal matrix acting as multiplicative identity).
Shows how matrices represent systems of linear equations.
A system Ax=b can be represented compactly using matrix multiplication, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector.
Introduces invertible matrices and the concept of matrix inverse.
A square matrix A is invertible if an inverse A⁻¹ exists such that AA⁻¹ = A⁻¹A = I (identity matrix). Not all matrices are invertible (e.g., zero matrix).
Defines the determinant for 2x2 matrices and its role in invertibility.
For a 2x2 matrix [ [a, b], [c, d] ], the determinant is ad-bc. A 2x2 matrix is invertible if and only if its determinant is non-zero. Determinants generalize to nxn matrices and determine invertibility.
Introduces notations for sets of matrices and groups.
M<0xE2><0x82><0x99>(F) denotes the set of nxn matrices with entries from field F. GL<0xE2><0x82><0x99>(F) denotes the general linear group (invertible matrices). SL<0xE2><0x82><0x99>(F) denotes the special linear group (matrices with determinant 1).
Explains vector spaces as fundamental algebraic structures.
A vector space V over a field F is a set with addition and scalar multiplication satisfying eight axioms (commutativity, associativity, identity, inverse for addition; distributive laws, scalar identity, scalar associativity).
Defines subspaces as smaller vector spaces within a larger one.
A subspace W of V must itself be a vector space under the same operations and contain the zero vector.
Introduces linear combinations, span, and linear independence.
Linear combination: a₁v₁ + ... + a<0xE2><0x82><0x99>v<0xE2><0x82><0x99>. Span(S): the set of all linear combinations of vectors in S. Linearly independent: only the trivial solution (all coefficients zero) to a₁v₁ + ... + a<0xE2><0x82><0x99>v<0xE2><0x82><0x99> = 0 exists.
Defines basis and dimension of a vector space.
A basis is a minimal set of vectors that spans the entire vector space and is linearly independent. The dimension is the number of vectors in any basis.
Explains linear transformations and their representation by matrices.
A linear transformation T: V -> W is a function that preserves vector space operations (T(cx+y) = cT(x) + T(y)). Once bases are chosen for V and W, T can be uniquely represented by an m x n matrix.
Introduces eigenvalues and eigenvectors.
An eigenvector v of a linear transformation T (or matrix A) is a non-zero vector such that T(v) = λv (or Av = λv) for some scalar λ, called the eigenvalue. The direction of the eigenvector remains unchanged (only scaled).
Explains how to find eigenvalues using the characteristic polynomial.
Eigenvalues λ are found by solving det(A - λI) = 0, where det is the determinant and I is the identity matrix. The polynomial det(A - λI) is the characteristic polynomial.
Defines trace of a matrix as the sum of its diagonal entries.
Trace(A) = Σ Aii. Trace is invariant under similarity transformations (Trace(A) = Trace(P⁻¹AP)).
Explains similar matrices and their shared properties.
Matrices A and B are similar if B = P⁻¹AP for some invertible matrix P. Similar matrices represent the same linear transformation in different bases and share properties like determinant, trace, and eigenvalues.
Introduces inner product spaces and orthogonality.
An inner product generalizes the dot product, allowing geometric concepts like length (norm = sqrt(<v,v>)) and orthogonality (<v,w> = 0) in abstract vector spaces. Satisfies linearity, conjugate symmetry, and positive definiteness.
Defines orthogonal and orthonormal bases.
An orthonormal basis consists of mutually orthogonal vectors (inner product is zero) each with a norm of one.
Chapter 3: Abstract Algebra
Defines binary operations and group axioms.
A binary operation '*' on set G is associative if (a*b)*c = a*(b*c). A group requires associativity, an identity element (e*a = a*e = a), and an inverse for each element (a*a⁻¹ = a⁻¹*a = e). An Abelian group is commutative (a*b = b*a).
Provides examples of groups.
The integers under addition (Z, +), rationals under addition (Q, +), reals under addition (R, +), complex numbers under addition (C, +), and Zn under addition modulo n (Zn, +) are groups. Q under multiplication is not a group because 0 has no inverse.
Explains subgroups as smaller groups within a larger group.
A subset H of a group G is a subgroup if H is itself a group under G's operation. Examples: Z is a subgroup of Q; SL₂(R) is a subgroup of GL₂(R).
Defines cyclic groups generated by a single element.
A group is cyclic if all its elements can be generated by repeatedly applying the group operation to a single element (the generator). Examples: Z (generated by 1 or -1), Zn (generated by 1).
Introduces group isomorphism: groups with the same structure.
Two groups (G, *) and (G', *') are isomorphic if there's a bijective map (one-to-one and onto) preserving the operation (φ(a*b) = φ(a)*'φ(b)). Isomorphic groups have the same structure. Example: Z₃ isomers to μ₃ (cube roots of unity under multiplication).
Defines rings as structures with two operations (addition and multiplication).
A ring (R, +, *) requires (R, +) to be an Abelian group, multiplication to be associative, and distributive laws to hold (a*(b+c) = a*b + a*c). Examples: Z, Q, R, C, M<0xE2><0x82><0x99>(R), Z<0xE2><0x82><0x99>.
Defines characteristic of a ring.
The characteristic is the smallest positive integer n such that na = 0 for all a in the ring. If no such n exists, the characteristic is 0. Example: Char(Z<0xE2><0x82><0x99>) = n; Char(Z) = 0.
Introduces integral domains and fields.
An integral domain is a commutative ring with unity and no zero divisors (ab=0 implies a=0 or b=0). A field is a commutative ring with unity where every non-zero element is a unit (has a multiplicative inverse). All fields are integral domains.
Defines ideals as special subsets for constructing quotient rings.
An ideal I in a ring R is an additive subgroup such that for any r in R and i in I, ri and ir are in I. Ideals are necessary for forming factor rings (R/I).
Defines factor rings (quotient rings) using ideals.
R/I is the set of cosets {r+I | r ∈ R}, forming a ring under induced operations. R/I is a field if and only if I is a maximal ideal in a commutative ring R with unity. R/pZ is a field for prime p.
Explains polynomial rings and irreducibility.
F[x] is the ring of polynomials with coefficients in field F. A polynomial is irreducible over F if it cannot be factored into lower-degree non-constant polynomials in F[x].
Generalizes divisibility and introduces prime/irreducible elements in integral domains.
Divisibility (a|b if b=ac) extends to rings. Prime elements satisfy: p|ab implies p|a or p|b. Irreducible elements cannot be factored non-trivially (p=ab implies a or b is a unit). In UFDs, prime and irreducible are equivalent.
Defines Unique Factorization Domains (UFDs).
An integral domain where every non-zero non-unit element has a unique factorization into irreducible elements (up to order and units). Examples: Z, F[x] for field F. Z[√-5] is NOT a UFD.
Introduces Eisenstein integers and their properties.
Z[ω] (where ω = e^(2πi/3)) is the ring of Eisenstein integers. It's a UFD and has a norm function (N(a+bω) = a²-ab+b²) useful for proving lemmas related to Fermat's Last Theorem.
Defines field extensions and their degree.
E over F (E⊃F) is a field extension. The degree [E:F] is the dimension of E as a vector space over F. Degrees are multiplicative: [E:K][K:F] = [E:F].
Defines algebraic elements and algebraic extensions.
An element α is algebraic over F if it's a root of a non-zero polynomial in F[x]. An extension E/F is algebraic if all elements of E are algebraic over F.
Defines algebraically closed fields.
A field K is algebraically closed if every non-constant polynomial in K[x] has a root in K. Example: C is algebraically closed; R and Q are not.
Introduces algebraic closure of a field.
An algebraic closure F of F is the smallest algebraic extension that is algebraically closed. Every field has an algebraic closure. Example: C is the algebraic closure of R.
Explains field automorphisms fixing a base field.
An automorphism σ of E over F is an isomorphism E -> E that fixes every element of F (σ(a)=a for a∈F). These form the Galois group Gal(E/F).
Introduces the absolute Galois group of Q.
Gal(Q̄/Q) is the group of automorphisms of the algebraic closure of Q that fix Q. It's a vast, complex group central to modern number theory, studied via its actions on modules (Galois representations).
Defines modules as generalizations of vector spaces.
An R-module is an Abelian group with scalar multiplication by elements from a ring R, satisfying similar axioms to vector spaces. Key difference: scalars come from a ring, not necessarily a field.
Distinguishes modules from vector spaces regarding bases.
Modules don't always have bases (e.g., Q as a Z-module), while vector spaces over fields do. Modules with bases are called free modules.
Explains Galois representations translating Galois group actions to module automorphisms.
A Galois representation maps elements of the absolute Galois group (e.g., Gal(Q̄/Q)) to automorphisms of a module (e.g., Tate module of an elliptic curve). This makes the abstract Galois group's actions more concrete via matrices.
Defines irreducible Galois representations.
An irreducible representation cannot be decomposed into smaller, stable representations acting on proper subspaces. This property is crucial for criteria like the Ribet's theorem.
Elliptic Curves
Projective plane provides a cleaner geometric setting than affine plane.
The projective plane (P²), unlike the affine plane (A²), ensures that any two distinct lines intersect at exactly one point, including parallel lines meeting at points at infinity. Points are represented by homogeneous coordinates [x:y:z].
Curves in projective space are defined by homogeneous polynomial equations.
For a curve to be well-defined in projective space, its defining polynomial must be homogeneous (all terms having the same degree).
Distinguishes between smooth and singular points on curves.
A point on a curve is singular if all partial derivatives of its defining polynomial vanish there. Curves without singular points are smooth; otherwise, they are singular. Elliptic curves must be smooth.
Defines an elliptic curve as a smooth cubic curve with a point at infinity.
An elliptic curve is a smooth curve defined by a homogeneous cubic equation in projective space that possesses at least one rational point (including the point at infinity).
Elliptic curve points form an Abelian group with geometric addition.
Points on an elliptic curve form an Abelian group under a geometric addition law: draw a line through P and Q, find the third intersection R, reflect R through the x-axis (for standard forms) to get P+Q. The point at infinity acts as the identity.
Reduction modulo p maps elliptic curves to finite fields.
Reducing the coefficients of an elliptic curve's equation modulo a prime p transforms it into a curve over the finite field Fp (or Fp^n). This finite set of points allows for counting.
Reduction modulo p can change a curve's smoothness (good vs. bad reduction).
An elliptic curve has 'good' reduction at prime p if the reduced curve is smooth. It has 'bad' reduction if the reduced curve has singular points (multiplicative or additive reduction). This is determined by the discriminant and C4.
Hasse's Theorem bounds the number of points on an elliptic curve over Fp.
For an elliptic curve E over Fp, the number of points |E(Fp)| is close to p+1, specifically | |E(Fp)| - (p+1) | ≤ 2√p. The difference is bounded by 2√p.
n-torsion points form a group isomorphic to (Z/nZ)².
The set of points P on an elliptic curve such that nP = O (the identity) forms a group E[n]. Over the algebraic closure, E[n] is isomorphic to (Z/nZ) × (Z/nZ).
Conductor of an elliptic curve measures its bad primes.
The conductor is an integer related to the primes where the elliptic curve has bad reduction, recording the powers of these primes. Minimal models simplify this analysis.
J-invariant classifies elliptic curves up to isomorphism.
The J-invariant is a number associated with an elliptic curve that remains invariant under isomorphism. Two elliptic curves are isomorphic if and only if they have the same J-invariant.
Frey curve construction links Fermat's equation to elliptic curves.
Assuming a non-trivial solution aᵖ + bᵖ + cᵖ = 0 leads to the Frey family of elliptic curves y² = x(x-aᵖ)(x+cᵖ). This construction is crucial for the proof.
Frey curves are semi-stable, implying modularity.
The Frey curve is proven to be semi-stable (good or multiplicative reduction at all primes). Its semi-stability guarantees, via the Modularity Theorem, that it corresponds to a modular form.
Modular Forms
Dedekind eta function and its transformation properties.
The Dedekind eta function η(τ) is defined via a product involving q = e^(2πiτ). It satisfies specific transformation laws under actions of SL₂(Z), which are cleaner when considering η(τ)²⁴.
Modular forms satisfy transformation laws under modular group actions.
A modular form of weight k for a group G satisfies f(γτ) = (cτ+d)<0xE2><0x81><0xBB>ᵏ f(τ) for γ = [[a,b],[c,d]] ∈ G. The modular group is typically SL₂(Z) or its congruence subgroups.
Eisenstein series are a specific type of modular form.
Eisenstein series E<0xE2><0x82><0x96>(τ) are defined as sums over divisors and satisfy modular transformation laws. They have a constant term (E<0xE2><0x82><0x96>(τ) starts with 1 in its q-expansion).
The upper half-plane (H) is the domain for modular forms.
H = {τ ∈ C | Im(τ) > 0}. This space is acted upon by the modular group.
The modular group SL₂(Z) acts on the upper half-plane.
The action is given by γ(τ) = (aτ+b)/(cτ+d) for γ=[[a,b],[c,d]]∈SL₂(Z). This action preserves the upper half-plane.
Fundamental domain captures each orbit exactly once.
A fundamental domain for the modular group action on H is a region containing exactly one representative from each orbit of the group action.
Congruence subgroups are important subsets of the modular group.
Defined by conditions on entries modulo n (e.g., Γ₀(n), Γ(n)). They play a crucial role in the theory, particularly related to elliptic curves.
Cusps are boundary points of the modular curve.
Rational numbers and infinity are considered cusps. Modular forms' behavior at these points (like vanishing) is important.
Modular forms are classified by weight and transformation properties.
Modular forms satisfy transformation laws under modular group actions. Cusp forms vanish at cusps, and holomorphic modular forms are finite at cusps.
Spaces of modular forms form vector spaces.
For a fixed weight k and group G, the set of modular forms M<0xE2><0x82><0x96>(G) and cusp forms S<0xE2><0x82><0x96>(G) form finite-dimensional complex vector spaces.
Peterson inner product defines geometric structure on the space of cusp forms.
An inner product allows notions of length and orthogonality for cusp forms, turning the space into a Hilbert space.
Hecke operators act linearly on spaces of modular forms.
Hecke operators T<0xE2><0x82><0x99> act on modular forms, shifting coefficients in their q-expansions. They commute and are self-adjoint with respect to the Peterson inner product.
Hecke forms are simultaneous eigenvectors of all Hecke operators.
A Hecke form is a modular form (often normalized) that is an eigenvector for every Hecke operator T<0xE2><0x82><0x99>. Its coefficients are multiplicative.
Eisenstein series are Hecke forms.
The Eisenstein series are examples of Hecke forms and form a basis for the space of modular forms when combined with cusp forms.
L-functions connect Hecke forms to number theoretic zeta functions.
Hecke forms have associated L-functions constructed from their Fourier coefficients, mirroring the structure of other number-theoretic L-functions like the Riemann zeta function.
Connecting Elliptic Curves and Modular Forms
Matching L-functions is a key clue suggesting a connection.
The L-functions derived from point counts of elliptic curves (over finite fields) and from the coefficients of Hecke forms exhibit identical structures and formulas, particularly for weight 2 modular forms.
Galois representations provide an algebraic link.
Both elliptic curves (via Tate module) and modular forms (via Hecke forms) can be used to construct Galois representations. The equivalence of these representations implies a deep connection.
The Modularity Theorem states the equivalence.
Every elliptic curve over Q is modular, meaning it corresponds to a weight 2 newform whose level matches the elliptic curve's conductor, and whose L-function/Galois representation matches that of the elliptic curve.
The J-invariant of an elliptic curve matches the modular J-function.
The J-invariant classifier for elliptic curves corresponds to the values of the modular J-function evaluated at specific points in the upper half-plane.
Frey curve's properties link to modularity.
The Frey curve, derived from a hypothetical Fermat solution, is semi-stable, guaranteeing its modularity. This connection is central to Wiles's proof strategy.
Ribet's Theorem connects Galois representations and modular forms.
Ribet's theorem implies that an irreducible mod p Galois representation arising from an elliptic curve must come from a modular form of the same level (conductor).
Proof of Fermat's Last Theorem
Assume a non-trivial solution 'aᵖ + bᵖ + cᵖ = 0' exists.
Starting with the assumption of a non-trivial primitive integer solution for Fermat's equation with prime exponent p ≥ 5.
Construct the Frey curve from the hypothetical solution.
The Frey curve y² = x(x-aᵖ)(x+cᵖ) is constructed based on the assumed solution. Properties of a, b, c ensure specific characteristics of this curve.
The Frey curve is semi-stable.
Analysis of the Frey curve's discriminant and related quantities shows it has good or multiplicative reduction at all primes, meaning it is semi-stable.
Semi-stability implies modularity by the Modularity Theorem.
Since the Frey curve is semi-stable, the Modularity Theorem guarantees it corresponds to a weight 2 newform of level equal to the curve's conductor.
Ribet's Theorem connects the Frey curve's Galois representation to modular forms.
Ribet's Theorem states that if a mod p Galois representation associated with the Frey curve is modular (which it is due to semi-stability), it must come from a modular form of level 2.
Contradiction arises from the level of the modular form.
The Frey curve's conductor determines the level of the corresponding modular form. Analysis shows this conductor must be 2. However, the structure of the Frey curve implies its corresponding modular form should have a level related to 'abc', which cannot be 2.
The contradiction proves the initial assumption is false.
The existence of a modular form at level 2 contradicts the properties derived from the hypothetical Fermat solution (which implies a higher level). Therefore, the initial assumption of a non-trivial solution must be false.
Fermat's Last Theorem is proven: no non-trivial integer solutions exist for n > 2.
The proof demonstrates that the chain of reasoning, starting from a hypothetical solution and leading to contradictions via modularity, confirms Fermat's Last Theorem.
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