Number Theory 2: Local Fields, spring 2015
Planning
There will be 12 lectures: 11:00-12:30 of every Monday from Feb. 16th to May 18th, except in April (with two lectures on Apr. 10th and Apr. 24th, both Friday).
The lectures take place in Room 407, MI, Leiden University. Lectures, homework and exam will be in English.
Attention: There is no lecture on May 18th.
Prerequisites
Algebraic Number Theory of the fall semester.
Course Description
In this course we will study local fields.
The main reference is the book "Local Fields (GTM 67)" by Jean-Pierre Serre. More concretely we will study
Basic facts and some ramification theory about local fields. For this part
we will mainly follow Chapter 7 of Milne's note (ANT),
with some extra materials from Part 1 and Part 2 of Serre's book.
The cohomology of profinite groups following Chapter 7 and 8 of Serre's book. Some sections (e.g. 7.8, 8.4 and 8.5) will be omitted and some extra materials
about cohomology of profinite groups will be added. Another useful reference for this part
is Chapter 2 of Milne's note (CFT). At the end of this part, we will treat
the theorem of Tate and Nakayama covered by Chapter 9 of Serre's book.
The Galois cohomology and the local class field theory. The main reference for this part will be (part of)
Chapter 10, 12 and 13 of Serre's book. Chapter 3 of Milne's note (CFT)
is also helpful.
The local Tate duality. We will start by discussing about cohomology dimensions and then define the dualizing modules to get the local Tate duality theorem. Click here for the course note of this chapter.
Homework
Since this is a master course, solutions to the homework sets need to be tex-ed. The teaching assistant of this course is Carlo Pagano.
Please send your solutions (in .pdf form) to carlein90 AT gmail.com (cc to gaoz AT math.leidenuniv.nl) by the due time of each homework set (see below).
The solutions to the homework exercises will be graded ++ (very good), + (good), +- (not so good) or - (poor).
Exam
The course is worth 6 EC. There will be a written exam. The final grade (a half-integer between 1 and 10) is meant to reflect the student's performance on the semester's homework exercises and the written exam.
More concretely, the final grade will be Max((homework+exam)/2, exam). Only tex-ed homework will be taken into account in this formula.
The exam will last 3 hours. You can take the following documents during the exam: course notes, Milne's online note Algebraic Number Theory, Serre's book "Local Fields", exercise sheets and your solutions. All results stated and proved in the course and in the exercise sheets can be directly used during the exam.
The exam consists of several exercises similar to the exercises in the homework. It will be at 14:00-17:00 on Wednesday 24 June at Room 174.