Math 18.786. Algebraic Number Theory (Fall 2011)
Instructor: Sug Woo Shin (swshin + at + institution name
dot edu)
Class: TR 11-12:30, Office hours: M 10-11, R 2-3
*** NOTE the change: M 11-12 is now moved to M 10-11 due
to conflict with
grad
student lunch seminar and D.W. Weeks lecture series
(the two seminars alternate) ***
(interested undergrads would also benefit from the
seminar;
female students are especially encouraged to attend the
Weeks lecture series; FREE pizza at the end of either seminar)
Neu=Neukirch Algebraic Number Theory, SD=Swinnerton-Dyer A Brief Guide to Algebraic Number Theory
1. Sep
08: Introduction/motivation, integrality, integral closure, ring of integers
(reading assignment by Sep 13:
Atiyah-MacDonald [Intro to commutative algebra]: Ch 1, Ch 2, Ch 5 pp.59-61, Ch
7 pp.80-82, Ch 9)
2. Sep
13: traces, norms, discriminants, definition of Dedekind domains
(reading: Neu I.2 or SD pp.1-9)
3. Sep
15: fractional ideals, unique factorization, ideal class groups [addendum]
(reading: Neu I.3 or SD pp. 9-15)
4. Sep
20: lattices
(reading: Neu I.4)
5. Sep
22: Minkowski theory, finiteness of class numbers [addendum]
(reading: Neu I.5-I.6 or SD pp.18-20)
6. Sep
27: Dirichlet Unit theorem [addendum]
(reading: Neu I.7 or SD pp.21-23)
7. Sep
29: proof of Unit theorem; primes in field extensions (change of fields)
(reading: Neu I.8 or SD pp. 25-26)
8. Oct
04: primes in Galois extensions, decomposition groups
(reading: Neu I.9 or SD pp. 26-30)
9. Oct
06: Inertia groups and Frobenius elements [addendum] –
a solution of HW #3, 6 and more
(reading: same as #8; for Columbus day
fun, read Neu I.11-14. If you have interest in algebraic geometry, you will be
delighted by I.13-14.)
10. Oct
11: MIT holiday
11. Oct
13: Cyclotomic fields, direct/inverse limits
(reading: Neu I.10 or Milne ANT Ch.6
pp.91-96)
12. Oct
18: intro to valuations
(reading: Neu II.1-3 or Milne ANT Ch.7
pp.101-106)
13. Oct
20: completions, Hensel’s lemma, extending valuations in finite extensions
(reading: Neu II.4 completions or Milne
ANT Ch.7 pp.110-119)
14. Oct
25: IN-CLASS MIDTERM EXAM (up to
cyclotomic fields) [Exam] [Sol
of some problems]
(in the solution, include p=2 if n=4, 6,
12 even if 2 is not 1 or -1 mod n; this comes from dealing with the possibility
that some p dividing n may be totally split in Q(zeta+zeta^{-1}) even if it is
ramified in Q(zeta).)
15. Oct
27: valuations in finite extensions of complete fields and number fields;
Galois orbit-place correspondence
(reading: Neu II.4, pp.131-133, II.8,
pp.160-163)
Note: Notice that Neukirch does not
assume that the valuation is discrete.
(For instance this is the case for the algebraic closure of Q_p.) Then the
valuation ring is not a DVR and the formalism of Dedekind domains does not
apply directly. This forces him to come up with more complicated proofs.
16. Nov 01: local and global Galois
extensions; lattices and chi-functions
(reading: Neu II.9-10 ;Serre III.1)
17. Nov 03: discriminant and different [addendum]
(reading: Serre III. 2-4, Neu III.2)
18. Nov 08: ramification criterion via
different; tame/wild ramification; computation of different
(reading: Serre III. 5-6)
19. Nov 10: intro to ramification groups
(reading: Serre IV. 1)
20. Nov 15: More on ramification groups [addendum]
(reading: Serre IV. 2)
21. Nov 17: Functions phi and psi; Herbrand
theorem
(reading: Serre IV. 3)
22. Nov 22: Hasse-Arf theorem
(reading: Serre V or a somewhat different
[Y] section 6; I followed the latter. The key case G=G_1=cyclic is treated in
Prop 6.6 of [Y])
*** There are two popular (purely local) approaches to
local class field theory (LCFT) these days. One is via Galois cohomology – this
has the benefit that the same formalism largely applies to global class field
theory (GCFT) as well but requires additional work on the so-called existence
theorem in LCFT. The other is via Lubin-Tate formal groups, which is very
efficient in the local setting but whose imitation in the global setup does not
go too far in establishing GCFT. We will take the latter approach. Main
references are as follows.
[LT] J. Lubin and J. Tate, Formal Complex Multiplication in Local
Fields, Annals of Math 81, 1965, pp. 380-387
[Serre] J.-P. Serre (esp. Chapter 3), Local
Class Field Theory, in [CF67]
[Y] T. Yoshida, Local Class
Field Theory via Lubin-Tate Theory, Ann. Fac. Sci. Toulouse Math 17, 2008,
pp. 411-438
23. Nov 29: Intro to LCFT, preliminaries
(maximal unramified/tame ext; completed ext; Weil groups)
(reading: Neu IV.1, [Y] sec 2)
24. Dec 01: Main theorems of LCFT, Lubin-Tate
formal groups
(reading: [Y] sec 3)
25. Dec 06: Lubin-Tate extensions (totally
ramified case) [addendum]
(reading: [Y] sec 4.1)
- Although
our setup looks like a bunch of algebra, the general picture is more geometric.
A satisfactory approach would be to study the deformation (or lifting) space of
formal modules from char p (like F_p) to char 0 (like Z_p or Q_p) through tools
in p-adic geometry, e.g. rigid geometry. The general GL(n) case (ours being
GL(1) case) is referred to as non-abelian Lubin-Tate theory as it is concerned
with full algebraic extensions (rather than abelian ones) of p-adic fields. I
know of few easy expositions from the geometric viewpoint which do not assume
much background – in fact [Y] itself is a good introduction in that it is
exactly written with a view toward generalization and the general geometric
picture. Maybe recent
lecture notes by Weinstein are still somewhat accessible.
26. Dec 08: Lubin-Tate extensions (totally
ramified + unramified extensions)
(reading: [Y] sec 4.2)
27. Dec 13 (Last Day): Proof of main theorems
of LCFT (omitting the proof of local base change, [Y] Thm 5.15) [proof of uniqueness]
(reading: [Y] sec 5 and 6.3)
Homework #1 (due Sep 20) [PDF]
Homework #2 (due Sep 27) [PDF]
Homework #3 (due Oct 04) [PDF]
(revised: Sep 29)
Homework #4 (due Oct 13) [PDF] (Oct
11 is a holiday)
Homework #5 (due Oct 25) [PDF] (solution to problem 4 by Holden Lee)
Homework #6 (due Nov 08) [PDF]
Homework #7 (due Nov 15) [PDF] (solution to problem 6)
Homework #8 (due Nov 22) [PDF] problems are scanned here
Homework #8.5: (to amuse yourself during Thanksgiving)
Find a more direct proof of Hasse-Arf theorem in G=G_1=cyclic case.
Final Homework #9 (due Dec 08) [PDF]