Math 689

Special Topics in Number Theory - Fermat's Last Theorem

1. History of Fermat's Last Theorem

   We will trace the history of Fermat's Last Theorem beginning with its pre-history in the study of Pythagorean triples and continuing through the the ideas of Diophantus of Alexandria to Fermat himself. We will give Euler's proofs of Fermat's Last Theorem for the cases n=3,4

2. Introduction to Number Theory
   1. Congruences
   2. Quadratic Reciprocity a la Gauss
3. Introduction to Elliptic Curves
   1. Plane cubic curves
   2. The group law on a plane cubic curve
   3. The projective plane
   4. Points of finite order and the Lutz-Nagell theorem
   5. The group of rational points and the Mordell-Weil Theorem
   6. Reduction mod p and cubic curves over finite fields
   7. The L-function of an elliptic curve and computations
4. Fermat's Last Theorem
   1. Cyclotomic fields
   2. Bernoulli numbers and regular primes
   3. Kummer's proof of Fermat for regular primes
5. Modular functions and Modular forms
   1. Definitions
   2. Examples and computations
   3. The "L-function" of a modular form
6. Modern developments and the proof of Fermats Last Theorem
   1. Frey curves
   2. Statement of the Shimura, Taniyama, Weil Conjecture (STW)
   3. STW implies Fermat and the work of A. Wiles.

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