SCHEDULE
The indicated paragraphs in [Gee] are often discussed in greater detail; e.g. some of the basic lemmas are proved rather than black-boxed. For a list of references, click on References above.
| Date | Topic | Reference | Remark |
|---|---|---|---|
| 02-04 | topology on Galois groups, definition/examples of Galois representations | [Gee] 2.1 and more | First Day |
| 02-06 | lemmas about l-adic reps, Brauer-Nesbitt theorem | [Gee] 2.3-2.5 | |
| 02-11 | number fields, local fields, Frob conjugacy classes, Chebotarev density thm | [Neu] II.9, [Gee] 2.24-2.25 | |
| 02-13 | local Galois reps (l not equal to p): Grothendieck's l-adic monodromy thm and Weil-Deligne reps | [Gee] 2.6-2.19 | |
| 02-18 | class holiday | reading assignment: local Galois reps when l=p; read the two pages from [Gee] 2.20 | Monday schedule of classes to be held. |
| 02-20 | Guest Lecture by Julee Kim (Topic: Intro to representation theory of GL(n) over p-adic fields) | See below if you wish to do some preparatory reading.* | SWS away to a workshop at MSRI |
| 02-25 | Galois deformations: group-theoretic hypothesis and universal lifting | [Gee] 3.1-3.3 | For motivation, read earlier sections of [Maz97]. |
| 02-27 | universal deformation, linear algebraic lemma | [Gee] 3.4-3.9 | |
| 03-04 | tangent spaces, generators and relations for universal lifting rings | [Gee] 3.10-3.14 | Review group cohomology before coming (minimum: [Ser-LF] VII.1-VII.3 or [AW] 1-2) |
| 03-06 | deformation conditions, generic fibers of universal lifting rings | [Gee] 3.15-3.17, [Stanford-4] | |
| 03-11 | Global deformation rings | [Gee] 3.18-3.22, just entered 3.23 | |
| 03-13 | Presenting global R over R^{loc}, computing H^i_{S,T}. | [Gee] 3.23-3.24 | See [Mil-ADT] 1.12, 2.3, 2.8, 4.10, 4.15, 5.1 or [Ser-GC] for facts on Galois cohomology |
| 03-18 | Dim and irred. components of local lifting rings when ell is not p | [Gee] 3.29-3.31 (cf. [BLGGT] 1.2-1.3), [Pil] sec 4 | Arizona Winter School |
| 03-20 | Local Galois reps and lifting rings when ell=p | [Gee] 3.27-3.28, 3.35-3.38 (cf. [CHT08] 2.4.1, [BLGGT] 1.4) | Review [Gee] 2.20-2.23 before coming; [Ber04] is a good survey on local Galois reps when ell=p |
| 03-25 | spring break | If you have time, preview [Gee] section 4. | Also review modular forms and auto. forms on GL(2), following articles in [FLT-white], [FLT-yellow], Bump's book, or Milne's course notes. |
| 03-27 | spring break | ||
| 04-01 | Basic def. in rep theory of p-adic groups, | [Gee] 4.1-4.3, either [BH] chap 1 or [Cas] sec 2 | |
| 04-03 | parabolic induction, irred. adm. in terms of supercuspidals, local Langlands for GL(n) | [Gee] 4.4-4.5, [Kud94] | |
| 04-08 | unramified local Langlands, Hecke operators, local base change | [Gee] 4.6-4.7 | For ref, [AC89] for base change, [Car79] sec 4.2 and [Bor79] sec 10.4 for Satake isomorphisms and unramified LLC in general. |
| 04-10 | local Jacquet-Langlands, automorphic forms on quat. alg. | [Gee] 4.8-4.14 | see [Rog83][DKV84] for local JL, [Gro99] for aut. forms on groups "cpt at infty" |
| 04-15 | global JL, global base change, global Langlands for GL(2) | [Gee] 4.15-4.23 | see [Bad08] for general global JL; [AC89] Ch.3 for global BC |
| 04-17 | integral theory of automorphic forms | [Gee] 5.2-5.3 | |
| 04-22 | MIT holiday | Patriots day | |
| 04-24 | Minimal Automorphy Lifting Theorem | [Gee] 5.1, 5.5-5.9 | cf. [Gee] Thm 5.1, which is a "non-minimal" ALT; cf. [BLGGT] Thm 2.3, [Tho12] Thm 7.1 for minimal ALT in higher dim. |
| 04-29 | Patching 1 | [Gee] 5.5-5.9 | See [Kis05] 2.3 for comparison of original and modified patching arguments. |
| 05-01 | Patching 2, completion of proof of minimal ALT | [Gee] 5.5-5.9 | |
| 05-06 | class holiday | SWS away to a workshop in Barbados | |
| 05-08 | John Binder's guest lecture | Topic: Eichler-Shimura theory | This includes a construction of elliptic curves from certain cuspforms of weight 2, a crucial ingredient in the proof of FLT. |
| 05-13 | On FLT, part 1 | ||
| 05-15 | On FLT, part 2 | Last Day |
