# Math 204B - Local Fields

Course description: This is the second in a string of three courses, which is an introduction to algebraic and analytic number theory. In part II we will discuss the basics of local fields. We will start from scratch with absolute values on fields, completions, Ostrowski, Hensel's lemma, ramification, Witt vectors, and then move on to division algebras over local fields, Brauer groups, Hasse invariant, and local class field theory. Depending on enrollment we will end with 1-2 weeks of student seminars on formal groups and Lubin-Tate theory.

Instructor: Claus Sorensen, csorensen [at] ucsd [etcetera]. Office hours by appointment in APM 6151.

Lectures: MWF 2-2:50, in the math department basement APM B412. (Beginning Mon Jan 6th.)

Textbooks/notes:

• J. Neukirch, Algebraic Number Theory, Springer Berlin Heidelberg, 1999. (E-version available here.)
• F. Lorenz, Algebra Volume II: Fields with structure, algebras and advanced topics (Book) Springer, 2008 (Universitext).
• J.-P. Serre, Local fields (Book) Springer, 1979 (GTM).
• J. Milne, Algebraic Number Theory, Notes, v. 2020 (Univ. of Michigan)
• J. Milne, Class Field Theory, Notes, v. 2020 (Univ. of Michigan)

Prerequisites: Math 204A

Homework: There will be no assigned homework. You may find a compilation of optional exercises here. I recommend you do as many of them as possible, but it is not required. Homework will not be graded.

Grades: You should expect to give at least two one-hour lectures in the Lubin-Tate seminar at the end, and provide TeX notes for your two talks.

Schedule/notes:

Week 1: Lecture Notes (Lifting simple roots mod p, Z_p as an inverse limit, absolute values, non-archimedean |*|, equivalence, Ostrowski for Q.)

Week 2: Lecture Notes (Ostrowski for k(t), complete fields, completions, valuation groups, R is Dedekind, discrete valuations, local compactness.)

Week 3: Lecture Notes (Local fields, global fields, pi-adic Laurent series, normalized |*| and the Haar measure, Newton's method for complete K.)

Week 4: Lecture Notes (Henselian rings, Hensel's lemma, Teichmuller lift, Hensel-Kurschak, extensions of |*| on complete K, Ostrowski's 2nd theorem.)

Week 5: Lecture Notes (Extensions of |*| for incomplete K, tensoring with completion, Ostrowski for K/Q, inertia degree and ramification index.)

Week 6: Lecture Notes (Unramified, totally ramified, tamely ramified extensions, K^ur and K^tr, inertia, Weil group, tame/wild inertia.)

Week 7: Lecture Notes (Krasner's lemma, root exchange, local fields are completions, C_p alg. closed, higher ramification and higher units.)

Week 8: Lecture Notes (Higher units, size of K^x mod n-powers, norm groups are open, exp/log, m^r=U^r for r>e/(p-1), structure of U^1.)

Week 9: Lecture Notes (Local class field theory, the Artin map, reciprocity, norm groups, formal groups and Lubin-Tate theory -- overview.)

Student seminars/notes on formal groups and Lubin-Tate theory:

Week 10: Lecture Notes (Randy, Finn, Bharatha.)

Week 11: Lecture Notes (Randy, Finn, Bharatha.)