Fermat's Last Theorem

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Conjecture (TWS): For every elliptic curve E of conductor N, there is a (unique) newform f(z) = ∑ a n (f)q n ∈ H + (N) of level N such that (4) a p (E) = a p (f), for all primes p̸ | N. Theorem (Shimura, 1971). For each f ∈ H + (N) with integral Fourier coefficients there is an elliptic curve E (of conductor N) such that (4) holds. Theorem (Wiles, 1995). Conjecture (TWS) is true if N E is squarefree. 6
5. TWS ss ⇒ FLT Suppose FLT p is false: there exist a, b, c ∈ Z with abc ≠ 0 such that a p + b p = c p . We may suppose (w.l.o.g.) that 2|a and that p ≥ 5. Consider the elliptic curve called a Frey curve. Then: 1) ∆ = (abc) 2p y 2 z = x(x − a p z)(x + b p z), 2) N E is squarefree (since 16|a p ). Thus, by Wiles’s theorem, there is an f = f E ∈ H + (N E ) such that (4) holds. Claim: Such an f E does not exist! Theorem (“Lowering the Level” - Ribet, 1991) Suppose f = f E ∈ H + (N) is a newform of level N. For a fixed prime number p > 3 let M p denote the product of the prime numbers q > 2 such that p|expt q (∆ E ). Then there exists g ∈ H + (N/M p ) such that a n (g) ≡ a n (f) (mod p), for all n ≥ 1 with gcd(n, N) = 1. Conclusion. Apply this to f E as above. Then by 1) we obtain that M p = N 2 , so by Ribet’s theorem there is a newform g ∈ H + (2). But this is impossible since dim S(2) = 0. 7
Page 1 and 2: Fermat’s Last Theorem Early Histo
Page 3 and 4: 3. A Basic Principle Problem: Find
Page 5: 4. The TWS-Conjecture To state this

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