# Harbin Number Theory Mini-course: Iwasawa theory

In order to promote the construction and sustainable development of number theory in Harbin, and to strengthen exchanges and cooperation with number theory scholars at home and abroad, we hereby launch a series of mini-courses in number theory. We will invite senior experts in the field of number theory at home and abroad to give lectures on a series of basic theories in the field of number theory. Interested teachers and students are welcome to actively participate!

Organisers: Chunhui Liu, Chao Qin, Jun Wang, Yichao Zhang.

Lecturer: Professor Meng-Fai Lim (Central Normal University)
Teaching Assistants:Chao Qin, Jun Wang
Course Time: Aug 8th -- Aug 13th
Q&A Time: Aug 10th -- Aug 13th (9:30--11:30)
Location: Room 901, Science Building, HIT

### Lecture One: Aug 9th 14:00-17:00

We review preliminaries on the power series ring in one variable with coefficient in the $p$-adic ring of integers. Along the way, we establish the Weierstrass Preparation Lemma. We also describe the structure theory of finitely generated modules over such power series ring.

### Lecture Two: Aug 10th 14:00-17:00

We will prove the Iwasawa asymptotic class number formula building on the structure theory. If time permits, we shall say something on the Kummer theory over a cyclotomic $\mathbb{Z}_p$- extension.

### Lecture Three: Aug 11th 14:00-17:00

We review the notion of Dirichlet $L$-functions. This is followed a discussion of the construction of $p$-adic $L$-function attached to these Dirichlet $L$-functions.

### Lecture Four: Aug 12th 14:00-17:00

We formulate the Iwasawa Main Conjecture for totally real fields (which is now a theorem of Mazur-Wiles, Wiles). We then give a sketch on its applications to the study of relating special values of Dedekind zeta functions to higher $K$-groups.

### References

P. Bayer and J. Neukirch, On values of zeta functions and l-adic Euler characteristics. Invent. Math. 50 (1978/79), no. 1, 35-64.
K. Iwasawa, On $\mathbb{Z}_l$-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326.
J. Neukirch A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd edn., Grundlehren Math. Wiss. 323 (Springer-Verlag, Berlin, 2008).
L. C. Washington, Introduction to cyclotomic fields. Grad. Texts in Math. 83. Springer-Verlag, New York, 1997.
C. Weibel, The K-book. An introduction to algebraic K-theory. Grad. Stud. Math. 145. Amer. Math. Soc. Providence, RI, 2013.