In the Spring 2016 semester I organized meetings to answer questions and lecture on the background for Prof. Zhang's course on class field theory. We met every Friday 1:30-2:30 in Math 528.

**Resources:**

Local Fields: Here are some notes on local fields. I lectured on these during our first two meetings. These notes also contain useful references.

**Topics covered/schedule:**

Date | Topics covered | References |
---|---|---|

1/29 | Local fields |
Local Fields Sections 1-3 [Milne] Ch. 7 |

2/5 |
Ramification in local fields The p-adic exponential and logarithm Brief introduction to global fields |
Local Fields Sections 4-5 [Milne] Ch. 7 |

2/12 |
Definition of global field Rings of integers of number fields Discriminants Quadratic fields |
[Milne] Ch. 2, or [Marcus] Ch. 2. [Lang] Ch. 1 for commutative algebra background. |

2/19 |
Splitting of primes in number fields Galois theory and prime decomposition |
[Marcus] Ch. 3 and 4 [Milne] Ch. 3 |

2/26 |
Local fields from global fields Examples of studying global fields locally |
[Milne] Ch. 7 and 8 |

3/4 |
Adeles and ideles Adelic proof of the finiteness of the class number |
[CF] Ch. 2 [Lang] Ch. 7 |

3/11 |
Compactness properties of adeles and ideles Adelic proof of the unit theorem |
[CF] Ch. 2 [Lang] Ch. 7 |

3/18 | Spring break | |

3/25 |
Minkowski theory Classical proof of the finiteness of the class number |
[Milne] Ch. 4 [Marcus] Ch. 5 |

4/1 |
Tate's Thesis preparation Abstract harmonic analysis |
[RV] Ch. 1-3 |

4/8 |
Tate's Thesis I Local functional equation |
[RV] Ch. 7 [Lang] Ch. 14 [LCNT] Section 11 |

4/15 |
Tate's Thesis II Global analysis |
[RV] Ch. 7 [Lang] Ch. 14 [LCNT] Section 12 |

4/22 |
Tate's Thesis III Global functional equation |
[RV] Ch. 7 [Lang] Ch. 14 [LCNT] Section 13 |

4/29 |
The Riemann-Roch theorem for number fields Lecture by Pak-Hin Lee |
My paper |

**References:**

[Milne] Milne's notes on Algebraic Number Theory.

A good introduction to the subject.

[Marcus] Marcus, *Number Fields.*

This book is a nice introduction to, well, number fields. It is very readable, and the last chapter motivates class field theory nicely. The drawback is that the local and adelic theories are nowhere to be found in this book.

[Lang] Lang, *Algebraic Number Theory.*

Extensive, but perhaps not great for beginners. Also, class field theory is not done using cohomology here. The last part is a nice source for some important analytic aspects of the theory, including Tate's Thesis.

[CF] Cassels and Frohlich, * Algebraic Number Theory.*

Beautiful. Uses local fields and adeles heavily. The basics are covered very quickly, however. Perhaps it's a better resource for class field theory. This is where original Tate's Thesis was published (though 17 years after it was written.)

[Neukirch] Neukirch, *Algebraic Number Theory.*

A very extensive and geometric approach to algebraic number theory. It even contains what is essentially the 1-dimensional case of Arakelov Theory. The treatment of class field theory, known as "Abstract Class Field Theory", is due to Neukirch himself. It does not use cohomology.

[Serre] Serre, *Local Fields.*

It's a theorem, or something, that everything written by Serre is beautiful.

[RV] Ramakrishnan and Valenza, *Fourier Analysis on Number Fields.*

Very nice and complete introduction to Tate's Thesis, and to the adelic approach to number theory in general. Function fields and number fields are treated on an essentially equal footing here.