Topics in Algebraic Number Theory

This is the webpage for the course Topics in Algebraic Number Theory (4373TOANT), taught at Leiden in the spring of 2022 by Alex Braat, Peter Stevenhagen, and Jan Vonk. The course consist roughly of two parts:

• The foundations of the theory of local fields, following notes by Stevenhagen.
• An introduction to the theory of p-adic L-functions, following notes by Vonk.

Lectures and exercise classes both will be held in person, in the Mathematical Institute (Snellius building). Students from outside Leiden are welcome, and should register officially. If you would like to audit the course without registering, please contact the lecturers via email to be added to the mailing list. The weekly schedule for this course is:

Lectures (Stevenhagen/Vonk)	Snellius 408	Fridays 11:15 - 13:00
Exercises (Braat)	Snellius 408	Fridays 14:00 - 15:45

Registered students who are taking this course for credit are required to hand in weekly homework assignments, which count for 20% of the final grade. Every Friday after the lecture, a short list of problems will be announced on this webpage, two of which are due to be handed in by 11.59 pm the day before the next lecture. You have free choice which two problems from the list you decide to hand in for grading each week. Handing in your solutions can be done by sending an email to Alex Braat with your typeset solutions attached. There is a final oral exam whose outcome constitutes 80% of the overall grade.

Lectures

Below is the schedule for the talks, as well as the homework assignments.

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Date	Lecturer	Subject
11 Feb 2022	Jan Vonk	Motivation and overview / Valued fields
	-	No exercise class (Dies Natalis)
		Homework: Choose two from 1.2,1.4,1.5,1.6,1.7,1.8,1.9,1.10,1.12,1.13
18 Feb 2022	-	* Cancelled *
	-	* Cancelled *
		Homework: * Cancelled *
25 Feb 2022	Jan Vonk	Complete fields
	Alex Braat	Exercises chapter 1.
		Homework: Choose two from 1.14, 1.17, 1.18, 1.19, 1.20, 2.1, 2.2, 2.3, 2.9, 2.10
4 Mar 2022	Peter Stevenhagen	Local fields and Hensel's Lemma
	Alex Braat	Exercises chapter 1/2.
		Homework: Choose two from 2.5, 2.6, 2.11, 2.13, 2.14, 2.15, 2.16, 2.17, 2.18, 2.23
11 Mar 2022	Jan Vonk	Extensions of valuations
	Alex Braat	Exercises chapter 2.
		Homework: Choose two from 3.2, 3.5, 3.8, 3.12, 3.13, 3.14, 3.15, 3.17, 3.18, 3.19
18 Mar 2022	Peter Stevenhagen	Extensions of complete fields
	Alex Braat	Exercises chapter 3.
		Homework: Choose two from 4.3, 4.5, 4.6, 4.7, 4.8, 4.9, 4.12, 4.13, 4.14, 4.16
25 Mar 2022	Jan Vonk	Galois theory of valued fields
	Alex Braat	Exercises chapter 4.
		Homework: Choose two from 5.2, 5.3, 5.4, 5.7, 5.8, 5.9, 5.10, 5.11, 5.14, 5.15
1 Apr 2022	Peter Stevenhagen	The Kummer-Dedekind theorem
	Alex Braat	Exercises chapter 5.
		Homework: Choose one from Chapter 6
8 Apr 2022	Peter Stevenhagen	Kummer theory
	Alex Braat	Exercises chapter 6.
		Homework: (None, Easter break)
15 Apr 2022	Lepus Paschalis	No meeting.
	Lepus Paschalis	No meeting.
		Homework:
22 Apr 2022	Jan Vonk	Foundations of p-adic analysis
	Alex Braat	Selected exercises and examples
		Homework: Choose two from 1.1, 1.2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8
29 Apr 2022	Jan Vonk	Foundations of p-adic analysis
	Alex Braat	Exercises chapter 2.
		Homework: Choose two from 2.10, 2.11, 2.12, 2.13, 2.14, 2.15, 2.16, 3.1, 3.2, 3.3
6 May 2022	Jan Vonk	Measures and distributions
	Alex Braat	Exercises chapter 2.
		Homework: Choose two from 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12
13 May 2022	Jan Vonk	p-Adic L-functions
	Alex Braat	Exercises chapter 4.
		Homework: Choose two from chapter 4.
20 May 2022	Jan Vonk	p-Adic L-functions
	Alex Braat	Exercises chapter 4.
		.
27 May 2022	-	* No meeting (Hemelvaart) *
	-
		.
3 June 2022	Jan Vonk	p-Adic L-functions
	Peter Stevenhagen	Factorisation via local methods

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