The Harvard University Certificate of Distinction in Teaching from the Derek Bok Center for Teaching and Learning was awarded for this course. I thank the enthusiastic support of all participants!

This is a website for the seminar course **Automorphic Forms and Galois Representations** (2008, Fall); here are some materials for this course. This seminar course will be followed by Math 254 in Spring 2009, by Richard Taylor.

*------* time

*------* news (**Email me now to (my first name)@gmail.com to be put on the mailing list!**)

**(06/11)** The notes for unitary Shimura varieties...

[pdf] **Notes for Week 7**

**(30/10)** The notes for etale cohomology...

[pdf] **Notes for Week 6**

Looking for volunteers to speak on the following dates - tell me if you're interested:

I'll be away on: **Nov. 24** (Mon); **Nov. 26** (Wed); **Dec. 1** (Mon)

*------* syllabus (**Rough plan of the course**)

Idea is to explain the construction of Galois representations associated to automorphic representations for GL_2(Q), but with general totally real/CM field case and GL_n case in mind. As things can be ultimately reduced to the analysis of compact unitary Shimura varieties, we sometimes ``pretend" that modular curves are compact, and/or talk about compact unitary Shimura curves instead.

0. Motivation; Class field theory (GL_1 theory) etc.

1. Modular Forms and Automorphic Forms, Automorphic Representations on GL_2 [Notes]

2. Betti cohomology of Shimura varieties, Matsushima formula [Notes]

3. Moduli interpretation of Shimura varieties [Notes, Notes from David, Notes from Ana]

4-5. Schemes, Moduli functors, Representability of moduli problems [Notes]

6. Etale cohomology and Galois representations, Singular-etale comparison

7. Proper/smooth base change, vanishing cycles

8. Integral model of Shimura varieties; Igusa curves and Lubin-Tate spaces

9. Lubin-Tate spaces: Serre-Tate theorem, Deformation of Barsotti-Tate groups, Drinfeld level structures

10. Lubin-Tate spaces as Rapoport Zink spaces, Eichler-Shimura congruence

11. Counting points on Igusa curves, Honda-Tate theory

12. Lefschetz trace formula, Weil conjecture

A1. Local Langlands correspondence and its proof

A2. Theory over \bar{Q}_p, Galois representation at p=l

A3. CM theory and more on unitary groups over CM fields

*------* notes

**Week 12** (15/12)

**Week 11** (08/12, 10/12, 12/12)

**Week 10** (03/12, 05/12)

**Away: Kyoto** (away 24/11, away 26/11, away 01/12)

**Week 9** (17/11, 19/11, 21/11)

**Week 8** (10/11, 12/11, 14/11)

**Week 7** (03/11, 05/11, 07/11)

**Week 6** (27/10, 29/10, 31/10) --- [pdf] **Notes for Week 6** (updated 30/10)

**Week 4-5** (15/10, 17/10, 20/10, 22/10, 24/10) --- [pdf] **Notes for Week 4-5** (updated 30/10)

(25/10) Finally the notes are coming together -- the main ideas were all explained in the lectures, but presented in a confusing order. Now the notes have the correct logical order (it seems). (Moreover, the diagrams are looking better with help from Liang Xiao.) I think I can fill in the other proofs before we move on to the etale cohomology theory.

[pdf] Notes for Week 4-5

(24/10) I brought up the proof of finiteness of the functors to illustrate the idea of patching, but I misjudged the difficulty of doing the proof on the blackboard (as usual). Once you lay down everything in the right order, the proof _is_ simple, and I seem to have been miraculously avoiding to make wrong statements in the class, but you need to wait for the notes for the accurate proofs - still under construction, but take a look -

[pdf] Notes for Week 4-5

(21/10) The notes are going slowly because I'm modifying the basic definitions back and forth - there are still inaccuracies, but I guess it's better than nothing - it may help you see the basic idea behind the lectures:

[pdf] Notes for Week 4-5

(16/10) Slowly typing the notes again:

[pdf] Notes for Week 4-5

(15/10) Slides on schemes as functors etc by Paul Goerss, thanks to Mark Behrens.

(12/10) Back from Bonn - hope you enjoyed the guest lectures - I put up the notes from David and Ana here (may be incorporated into my notes in the future). I couldn't work much on the notes in Bonn, but finally saw how to fix the last part of Week 2 notes with help from Yoichi Mieda (Kyushu); I'll work on it shortly. I fixed some mistakes in Week 1/2 notes thanks to Hwajong Yoo (Berkeley).

**Away: Bonn** --- **Guest Lectures**:

**Oct. 6** (Mon): Around the spectral decomposition of L^2(G) --- by **Daniel Kane**.

**Oct. 8** (Wed): Unitary Shimura varities over CM fields, I --- by **David Geraghty** [Notes]

**Oct. 10** (Fri): Unitary Shimura varities over CM fields, II --- by **Ana Caraiani** [Notes].

**Week 3** (29/09, 01/10, 03/10) --- [pdf] **Notes for Week 3** (updated 03/10)

**(03/10)** I added the discussion of isogeny vs isomorphism in more detail in the notes. In the lecture I didn't emphasize enough that the isogeny framework frees us from choosing a maximal compact subgroup (or an adelic lattice). See you on Oct. 15 Wed. again, and meanwhile, enjoy the guest lectures!

[pdf] Notes for Week 3

(02/10) Some preview for tomorrow here. I updated the notes for Week 2 as well - seems OK up to p.16, from p.17 on take it with a grain of salt... Richard has suggested to me that we might replace the spectral theory by the holomorphic theory of BGG spectral sequence (Faltings), but I haven't looked into it yet.

[pdf] Notes for Week 3

(30/09) A nice article by J.Arthur was pointed out by Sug Woo Shin, as related to my notes for Week 2 (not complete yet; I'll keep on working). Meanwhile we'll proceed with the moduli interpretation: here are some preview of tomorrow's lecture.

[pdf] Notes for Week 3

(29/09) I typed the part where we trivialize the holomorphic vector bundles (I departed from the Harris-Taylor notation by using curly V for holomorphic vector bundles; I changed the line bundles in Week 1 into V as well). It is tricky to keep track of which equivariance you retain at each moment, but I hope my notes address that issue clearly. Proof of Kodaira-Spencer and computation of U-types will be coming. I believe there are not much black-boxes in the Week 2 notes apart from the spectral decomposition of L^2 (in the classical setting, it'll be treated in Daniel's talk next week; in the adelic setting, I still lack a good reference). Well, de Rham theorem, Hodge theory, Harish-Chandra's admissible (g,U)-modules, discrete series for GL_2, etc sound formidable, but at least we are not using very deep results here.

[pdf] Notes for Week 2

[pdf] Notes for Week 3

(28/09) I typed up the bare skeleton of the argument I inteded to make -- there are still details to be filled in. Okay, the material of Week 2 got a bit too advanced; I do want to stick to modular curves for most of the time to convey the intuition, which will be the case from Week 3 again. Meanwhile I'll try to make the notes as precise as possible.

[pdf] Notes for Week 2

**Week 2** (22/09, 24/09, 26/09) --- [pdf] **Notes for Week 2** (updated 12/10)

(27/09) Notes are coming slowly... translating the literature for semisimple groups to our reductive group setting. I wrote the part where you pull out the central characters -- now I have to put it back in after moving to the holomorphic setting...

[pdf] Notes for Week 2

(26/09) Sorry for another rather disorganized (just as I promised!) lecture; I'll work on my notes during the weekend. The argument I had in mind was: (i) Matsushima formula (real analytic) --> (ii) Computing the relevant (g,U)-cohomology, where you see the Hodge theory coming in when H is Hermitian symmetric --> (iii) Showing how it recovers the holomorphic theory, namely the Eichler-Shimura isom for GL_2 (modulo the cusp issue). I will hopefully write up the notes for this and move on to the moduli interpretation/C, which connects from the vector bundles that I discussed at the end of today's lecture.

(25/09) I updated some explanations on Hecke correspondences, and typed the part where the trivialization of the local system. Some more on Lie algebra cohomology and (g,U)-cohomology stuff too.

[pdf] Notes for Week 2

(24/09) It's really tough for me to lecture through the material of this week, but it's been very helpful for me - I hope you are finding the lectures ok. It was funny to hear Matsushima's formula reviewed again in Oda's talk in the afternoon! Here are some more notes.

[pdf] Notes for Week 2

(22/09) Hope we can go through the real/complex stuff without much trouble this week! I incorporated the central character part of today's lecture into the notes of Week 1 below. I started typing the new file too.

[pdf] Notes for Week 2

**Week 1** (15/09, 17/09, 19/09) --- [pdf] **Notes for Week 1** (updated 12/10)

(18/09) Now the problem with GL_2(Q) seems to be fixed. See you tomorrow!

[pdf] Notes for Week 1

(17/09) Thank you for your continued enthusiasm. I do appreciate your insistent questionings (which helps me a lot), and it's a pity that we don't have enough time to go through the details slowly, which is actually the style I prefer. But I planned out this course so that we can see how the things fit together in a bigger picture, so it is also important to move on quickly too - we'll see how we can manage this dilemma. I hope my notes will help clarify the issues. I partly typed up the notes for today; decided not to divide the notes by lectures for now, as it will have cross-references and I'm continuously revising the notes. I tried to answer the objections concerning the lattices - how does it look now? Also, I included somewhat detailed explanations on adelic GL_n, which might be of some help.

[pdf] Notes for Week 1

(15/09-II) I did type up the notes - I had some previous articles to cut and paste this time; please don't expect as much from the next time on, although I'll try my best. Comments, pointing out of typos/mistakes welcome.

[pdf] Notes for Lecture 1

(15/09) Thanks for coming everyone - I feel encouraged and reassured to have many of my mistakes corrected. I went through the introductory stuff quickly, but I will certainly slow down when we get to more exotic material. I'll put up the notes on the web soon. For people who are new to modular forms, Serre's "A Course in Arithmetic" is the canonical place to start. As for automorphic forms, things suddenly get non-trivial - I've checked them out of Cabot at the moment, but I found the following books useful - Cogdell-Kim-Murty "Automorphic L-functions", Bump "Automorphic Forms and Representations", and Borel "Automorphic Forms on SL_2(R)". And don't forget to check out Richard's expository article: Galois representations, by R. Taylor.

We defined the automorphic representations as representations of GL_2(A^\infty), but there is a canonical way to think about automorphic representations as representations on GL_2(A), including GL_2(R). This is done by thinking of holomorphy and growth conditions near cusps of automorphic forms in a real analytic way, and constructing an admissible (g, K)-module or even unitary representation of GL_2(R) at infinity in the space of those automorphic forms. As we are sticking to "discrete series at R" case, we adopted a rather "geometric" definition. For GL_n where n is bigger than 2, there is no discrete series of GL_n(R) and this geometric definition wouldn't work any more (there is no reasonable Shimura variety for them either), but the real analytic definition of automorphic/cusp forms generalize.

Automorphic representations are the representations appearing in the space of automorphic forms. Being representations, they are more mobile, and will appear in other spaces too, most notably in the cohomology of Shimura varieties.

(14/09) I'll start with a general introduction tomorrow, but haven't fixed the prerequisites etc. - as course should cover wide range of materials, everyone (including me) must be a beginner at one subject or another. My plan is to ask motivated grad students to write up the proofs of key theorems for us to share. We'll see how it goes. See you tomorrow!

Here are some stuff which may be of interest for the grad students:

This 100-page note starts "from the scratch" and proves up to the prime decomposition law in cyclotomic fields and quadratic reciprocity.

Completely revised recently.