Math 204B  Number Theory (UCSD and online, winter 2021)
This site will disappear someday, but has been archived here.
Course description:
This is the second in a series of three courses, which is an introduction to algebraic and analytic number theory.
Part A treated the basic properties of number fields:
their rings of integers, unique factorization and its failure, class numbers,
the Dirichlet unit theorem, splitting of primes, cyclotomic fields, and padic fields.
Part B will continue with these topics, plus an introduction to class field theory (the study of abelian extensions of number fields).
As in part A, there will be some emphasis on computational tools, particularly SageMath
and the LMFDB.
(In spring 2021, Claus Sorensen will teach Math 204C.)
Due to the COVID19 pandemic and UCSD campus regulations, this course will be offered in a fully remote format.
Lectures will be delivered live via Zoom, and also recorded for asynchronous viewing.
Office hours will be held via Zoom; I also plan to offer some inperson office hours as conditions permit.
I can only grant course credit to UCSD enrolled students. UCSD offers crossregistration for students from California community colleges and Cal State campuses, and concurrent enrollment for others (for a fee which I do not control).
This course will use Canvas in the following ways only.
 Grades will be posted to Canvas (and nowhere else).
 Zoom links will be posted to Canvas, and also to Zulip.
 Lecture videos will be available in Canvas, and also via the public links on this page.
Online epicourse:
As with 204A, I plan to run a parallel "epicourse" for the general public. This will include the following components.
(All times are local to San Diego, which is UTC8.)

I will be streaming lectures in real time using Zoom.
For each lecture, I will provide a link to my live
whiteboard on Miro, which enables students to see all of the boards at once.

I will record each lecture and post it to Mediaspace shortly afterward (typically within one hour).
I will also post the whiteboards as PDFs on this web site.

In some cases I may prepare lecture notes, which will be posted on this web site.

I will hold office hours on Zoom, at multiple times to provide reasonable options in all time zones.

I also plan to have inperson office hours on the UCSD campus, to which epicourse participants based in San Diego are also
welcome (subject to UCSD campus regulations, including use of face coverings). Please contact me directly if you wish to attend.

I will use Zulip for textbased communication with and among the class, including announcements, Zoom passcodes/direct links, and discussion of the course material and problem sets.

I will set up a project on CoCalc to facilitate use of software (especially SageMath)
for some units of the course, and to collect homework for feedback. A few lectures will be given as live demonstrations in CoCalc,
borrowing a format from my course Math 157.
Thomas Grubb has provided a short video introduction to CoCalc.
The epicourse will be entirely unofficial (and free of charge); for credit, one must take the official course (see above).
If you wish to participate in the epicourse and did not participate in fall 2020,
please fill out this Google Form. You should receive an invitation to join Zulip; this may take a day or two
depending on how often I check the form. (I will keep updating Zulip even after the course starts, in case you want to join late.)
If you participated in the epicourse in fall 2020, you do not need to sign up again.
Note that the lecture recordings, whiteboards, and lecture notes will be posted publicly here, so you do not need to join the epicourse to get those.
However, I hope the other interactions via Zoom and Zulip will add significant value, and encourage everyone following the lectures to join.
Environment: In both the course and the epicourse,
I aim to create a conducive learning environment for those who do not see themselves
reflected in the mathematical profession at present and/or have experienced systemic bias affecting their mathematical education.
I insist that all participants do their part to maintain this environment. I also aim to address accessibility issues as best I can; please let me know directly if this might affect you.
Instructor: Kiran Kedlaya,
kedlaya [at] ucsd [etcetera].
Lectures: MWF 1010:50am, via Zoom (meeting code 964 2065 5406). All lectures will be available for remote viewing both synchronously and
asynchronously. I aim to have each lecture posted within one hour of completion (this is limited by the speed of video processing).
Office hours: Unless otherwise specified, these are for both the UCSD course and the epicourse. Timings may be adjusted during the term.
 Online: MWF 11:0011:30am (meeting code 963 5862 4216). This does not apply on days with no lecture (see below).
 Online: MW 88:30pm (meeting code 963 5862 4216). This does not apply on days with no lecture (see below).
 Hybrid (UCSD only): Wednesday 3:154:15pm, locations to be announced on Zulip (or Zoom meeting code 943 1311 6169).
NonUCSD course or epicourse participants are welcome in person, but please contact me first.
Textbook:
Primarily Algebraic Number Theory (Springer) by J. Neukirch.
(UCSD affiliates can download the text for free via the UCSD VPN.)
For class field theory, I will also refer to my own notes on class field theory (available in PDF or HTML).
As a supplement I recommend Milne's notes
Algebraic Number Theory
and
Class Field Theory.
You may also want to check out
Atiyah and MacDonald, Introduction to Commutative Algebra;
Lang, Algebraic Number Theory;
FröhlichTaylor, Algebraic Number Theory;
CasselsFröhlich, Algebraic Number Theory;
Jarvis, Algebraic Number Theory (the Math 104A/B text);
or Janusz, Algebraic Number Fields.
Additional references to be added later.
Prerequisites:
Math 204A or permission of instructor. I will grant permission based on background in algebra (at least Math 100AC, i.e., groups, rings, fields, and Galois theory) and number theory (at the level of Math 104A and 104B as they were taught in 20192020).
Please do not request enrollment authorization without contacting me separately.
If you were not following Math 204A and are depending on your independent knowledge of that material, I would recommend viewing the last two lectures (December 9 and 11) of that course in advance. They serve as an introduction to the material we will be considering here.
Homework: Weekly problem sets (47 exercises), due on Thursdays (weeks 210).
Homework will be submitted online via CoCalc. You are welcome (and strongly encouraged) to collaborate on homework and/or use online resources, as long as you (a) write all solutions in your own words and (b) cite all sources and collaborators. (However, for best results I recommend trying the problems yourself first.)
The maximum homework score will be 5 (formerly 6) out of 9 complete homeworks (by percentage, combining partial scores across all submitted assignments.)
I am also offering the option to make up any one problem set by submitting a short writeup of a topic that is related to the class but not covered in lecture. Please contact me if you wish to exercise this option.
Since the homework policy is more generous than last quarter, I am going to be a bit less generous about extensions this term; please ask for one
in advance of the due date if you need it. This will allow me to open up homework discussion on Zulip in a more timely fashion.
Final exam: None. Disregard any information from the UCSD Registrar to the contrary.
Grading: 100% homework; see above.
Key dates:

First lecture: Monday, January 4.

University holidays: Monday, January 18; Monday, February 15.

Last lecture: Friday, March 12.
Assignments: The numbering continues that of 204A.

HW 9: PDF, source (due Thursday, January 14).

HW 10: PDF, source (due Thursday, January 21).

HW 11: PDF, source (due Thursday, January 28).

HW 12: PDF, source (due Thursday, February 4).

HW 13: PDF, source (due Thursday, February 11).

HW 14: PDF, source (due Thursday, February 18).

HW 15: PDF, source (due Thursday, February 25).

HW 16: PDF, source (due Thursday, March 4).

HW 17: PDF, source (due Thursday, March 11).
Topics by date (with videos, references, notes, and boards):
See also this page for the videos embedded as iframes,
or this playlist for all the videos together.

Jan 4 (M): Kummer theory and the KroneckerWeber theorem (video, boards, Miro live).
References: CFT notes 1.1, 1.2, 1.3. See also "From quadratic reciprocity to class field theory" (pdf).

Jan 6 (W): The Hilbert class field (video, boards, Miro live).
References: CFT notes 2.1.

Jan 8 (F): Generalized ideal class groups and the Artin reciprocity law (video, boards, Miro live).
References: CFT notes 2.2.

Jan 11 (M): The principal ideal theorem (video, boards, Miro live).
References: CFT notes 2.3.

Jan 13 (W): Zeta functions and the Chebotarev density theorem (video, boards, Miro live).
References: CFT notes 2.4.

Jan 15 (F): Cohomology of finite groups, I (video, boards, Miro live).
References: CFT notes 3.1.
Note: this lecture is rescheduled to Thursday, January 14 at 4pm.

No lecture on Monday, January 18.

Jan 20 (W): Cohomology of finite groups, II (video, boards, Miro live).
References: CFT notes 3.2.

Jan 22 (F): Extended functoriality; homology and Tate groups (video, boards, Miro live).
References: CFT notes 3.2, 3.3.

Jan 25 (M): Herbrand quotient; profinite groups (video, boards, Miro live).
References: CFT notes 3.4, 3.5.

Jan 27 (W): cohomology of profinite groups; overview of local class field theory (video, boards, Miro live).
References: CFT notes 3.5, 4.1.

Jan 29 (F): overview of local class field theory (video, boards, (Miro live).
References: CFT notes 4.1.

Feb 1 (M): cohomology of local fields (video, boards, Miro live).
References: CFT notes 4.2.

Feb 3 (W): cohomology of local fields; Tate's theorem (video, boards,
Miro live).
References: CFT notes 4.2, 4.3.

Feb 5 (F): local CFT via Tate's theorem (video,
boards, Miro live).
References: CFT notes 4.3.

Feb 8 (M): abstract CFT (video,
boards, Miro live).
References: CFT notes 5.1, 5.2.

Feb 10 (W): the abstract reciprocity map and reciprocity law (video,
boards, Miro live).
References: CFT notes 5.2, 5.3.

Feb 12 (F): the abstract reciprocity law; the filtration on a local Galois group (video,
boards, Miro live).
References: CFT notes 5.3, 4.4.

No lecture on Monday, February 15.

Feb 17 (W): adèles (video, boards, Miro live).
References: CFT notes 6.1.

Feb 19 (F): idèles and class groups (video, boards, Miro live).
References: CFT notes 6.2.

Feb 22 (M): adèles and idèles in field extensions; the theorems of adelic CFT (video, boards, Miro live).
References: CFT notes 6.3, 6.4.

Feb 24 (W): localglobal compatibility for the reciprocity law; overview of the proofs of global CFT (video, boards, Miro live).
References: CFT notes 6.4, 6.5.

Feb 26 (F): the First Inequality (video, boards, Miro live).
References: CFT notes 7.1.

Mar 1 (M): the Second Inequality: analytic proof (video, boards, Miro live).
References: CFT notes 7.2.

Mar 3 (W): the abstract reciprocity map; reductions for the existence theorem (video, boards, Miro live).
References: CFT notes 7.3, 7.4.

Mar 5 (F): the key case of the existence theorem; algebraic proof of the Second Inequality (video,
boards, Miro live).
References: CFT notes 7.4.

Mar 8 (M): localglobal compatibility (video,
boards, Miro live).
References: CFT notes 7.5.

Mar 10 (W): Brauer groups of number fields (video,
boards, Miro live).
References: CFT notes 7.6.

Mar 12 (F): preview of Math 204C: adelic Fourier analysis (video,
boards, Miro live).
References: CFT notes 6.6; see also CasselsFröhlich, chapter 15
and RamakrishnanValenza, Fourier Analysis on Number Fields (the Math 204C textbook).