MA 358A: Topics in Number Theory 3 (Iwasawa theory)
Credits: 3:0
Prerequisites :
- MA 213 : Algebra 2
- MA 313 : Algebraic Number Theory
- This course will use material from MA 358 : Topics in Number Theory 2 ($p$-adic $L$-functions) and should be taken concurrently.
This course is an introduction to classical Iwasawa theory, up to the proof of the Iwasawa main conjecture following Mazur and Wiles. We will begin with a review of results from algebraic number theory, class field theory etc. This will be followed by a study of $\mathbb{Z}_p$ extensions of number fields. We will then concentrate on the cyclotomic $\mathbb{Z}_p$ extensions of number fields. This will be followed by formulation of the Iwasawa main conjecture. For this part we need knowledge of $p$-adic $L$-functions. If time permits we will see Wiles’s proof of the Iwasawa main conjecture.
Suggested books and references:
- K. Iwasawa, On $\mathbb{Z}_l$-Extensions of Algebraic Number Fields, Annals of Mathematics Vol. 98, No. 2 (1973).
- R. Greenberg, Iwasawa Theory -- Past and Present, Advanced Studies in Pure Mathematics 30 (2001).
- P. Deligne, K. Ribet, Values of Abelian L-functions at Negative Integers over Totally Real Fields, Inventiones Mathematicae Vol. 59 (1980).
- A. Wiles, The Iwasawa Conjecture for Totally Real Fields, Annals of Mathematics Vol. 131, No. 3 (1990).