Math 204B - Number Theory

Course description: This is the second in a string of three courses, which is an introduction to algebraic and analytic number theory. Part A treated the basics of number fields (their rings of integers, failure of unique factorization, class numbers, the Dirichlet unit theorem, splitting of primes, cyclotomic fields, and more). In part B, we will focus on local fields and local properties of number fields (completions of number fields, finite extensions of local fields, ramification, different and discriminant, decomposition and inertia groups, basics of local class field theory). Part C will focus on zeta functions. The course will be relatively elementary; no extensive background needed. It should be of interest not just to number theorists, but to algebraic topologists etc. The plan is to cover chapter II (in full) and chapter III (as time permits) in Neukirch's book, and supplement with material from Milne's notes. You should have both.

Instructor: Kiran Kedlaya, kedlaya [at] ucsd [etcetera]. Office hours 2-3:30, or by appointment, in APM 7202.

Lectures: MWF 1-1:50, in APM 7421. There may be also be some makeup lectures to account for cancelled lectures (see announcements).

Textbook: Primarily Algebraic Number Theory (Springer) by J. Neukirch. As a supplement I recommend Milne's notes Algebraic Number Theory and Class Field Theory. You may also want to check out Lang, Algebraic Number Theory; Fröhlich-Taylor, Algebraic Number Theory; Cassels-Fröhlich, Algebraic Number Theory; or Janusz, Algebraic Number Fields. For local fields, the canonical reference is Local Fields I am also working on collating and revising my old class field theory lectures notes from a 2002 course at Berkeley; here is the current draft.

Prerequisites: Math 204A or equivalent (with instructor's permission).

Homework: Weekly problem sets (4-6 exercises), due on Wednesdays.

Final exam: Take-Home. Due Wednesday, March 18, 3 PM; must be submitted by email. (For best results, if you hand-write your exam, please use a scanner rather than the camera on your phone; the department has one available. Of course, you may also submit typed solutions.)

Grading: 50% Homework, 50% Final

Announcements:

First lecture: Monday, January 5.
Holidays this term: Monday, January 19; Monday, February 16.
Other dates with no lectures: February 2-6.
Makeup lecture: Monday, February 16.

Assignments:

HW 1 (Due Jan 14): pdf.
HW 2 (Due Jan 21): pdf. Also available on SageMathCloud!
HW 3 (Due Jan 28): pdf. No SMC component this week.
HW 4 (Due Feb 11): pdf.
HW 5 (Due Feb 18): pdf. No SMC component this week.
HW 6 (Due Feb 25): pdf.
HW 7 (Due Mar 4): pdf (corrected Mar 3). No SMC component this week, but please use SMC to submit any Sage work that constitutes part of your solutions.
HW 8 (Due Mar 11): pdf. No SMC component this week.

Topics covered so far (with references and notes; CFT = class field theory notes, see above):

Jan 5: the product formula for Q; the algebraic and analytic interpretations of Q_p (Neukirch II.1, notes on completions).
Jan 7: the product formula for number fields; completions of number fields
Jan 9: valuations, completions (Neukirch through II.4.5)
Jan 12: Hensel's lemma, extension of valuations (Neukirch II.4.6--4.8)
Jan 14: completeness of finite extensions, local fields (Neukirch II.4.9--5.2, notes on Hensel's lemma)
Jan 16: characterization of local fields (Neukirch II.5.2), Haar measure on locally compact fields
Jan 21: Newton polygons (Neukirch II.6)
Jan 19: no lecture (university holiday)
Jan 23: Henselian fields and the unique extension property, the fundamental identity (Neukirch II.6; notes on infinite extensions)
Jan 26: Unramified and tamely ramified extensions (Neukirch II.7)
Jan 28: Calculations for cyclotomic extensions (Neukirch II.7)
Jan 30: Extensions of valuations, relationship with field embeddings (Neukirch II.8.1)
Feb 2-6: no lectures (Kiran away)
Feb 9: Galois theory of valuations (Neukirch II.9)
Feb 11: continuation of Neukirch II.9
Feb 13: Higher ramification groups (Neukirch II.10; see also Serre, "Local Fields")
Feb 16: optional lecture on perfectoid fields (see this paper)
Feb 18: discriminant and different (Neukirch III.2).
Feb 20: continuation of Neukirch III.2
Feb 23: continuation of Neukirch III.2
Feb 25: Galois cohomology (CFT 8).
Mar 2: More Galois cohomology (CFT 9), homology (CFT 10).
Mar 4: Overview of local class field theory (CFT 12, plus some asides).
Mar 6: Extended functoriality (CFT 10), profinite groups (CFT 11).
Mar 9: Cohomology of local fields (CFT 13)