This is a website for the course **Non-abelian Lubin-Tate Theory** (2007, M24); here are some material for this course.

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**Week 4** (29/10/07 - 02/11/07)

(02/11) Hi - sorry for all the mess I made this week. The colloquim talk took a heavy toll on my energy and I wasn't functioning well most of the week; finally catching up. First, here are the much-awaited notes. Sorry for the delay!

Let me apologize for the unclear lecture on Wednesday. I didn't exactly tell lies (at least, not big ones), but the way I presented the proof was very unfriendly. The part explaining the relative representability was okay, but in the proof of latter part I used many facts from commutative algebra without notice... especially unpardonable was the fact that I was using a version of **Weierstrass preparation**, a sort of division by formal power series over complete local ring. I wrote up my proof in the Appendix of the notes. There are three (more or less standard) facts from commutative algebra that I used without warning - (1) Krull's intersection theorem - the intersection of all powers m^r of maximal ideal in a noetherian local ring is trivial (an application of Artin-Rees lemma), (2) regular local ring is a domain (an application of some dimension theory), (3) finite flat A-algebra has same dimension as A (this might be the easiest). I hope you don't have too much difficulty in looking up the literature for proof of these. At least I should have isolated these facts before giving the lecture, and if you couldn't follow Lecture 11 that was my fault, not yours. Try looking up the notes when you can.

Let me explain where we are. Now we've gone through the first half of the term - I'm a bit behind than my original plan, but not too much. We've defined the basic object of study, the Lubin-Tate tower (bunch of n-dmiensional regular local rings, after all), and groups acting on it. So far we didn't have to use anything apart from what is in my prerequisite notes + some commutative algebra (but not too hard-core ones). And a rough form of **Non-abelian Lubin-Tate theory** is now easily stated - take the l-adic etale cohomology group of (the geometric generic fiber of) these spaces, and you get the **Jacquet-Langlands** and **local Langlands correspondence** between the representation of 3 groups - **the multiplicative group of the division algebra, GL_n(K), and the Galois group of K**. Now to make this statement into a precise form, you need to know A LOT more than what we've been using, and that is what I'll try to lecture in the latter half of the course. On the other hand, there is a basic question - **can we prove non-abelian Lubin-Tate theory locally** (i.e., without using the theory of Shimura varieties over global fields)? But if you think about this question carefully, the question isn't as well defined as it seems - although we do have a characterization of these correspondences, now that we know that there is this beautiful canonical object that realizes these correspondences, we are tempted to **define** these correspondence via Lubin-Tate tower, in the way we defined the Artin map in the abelian case. Then the question is to find a good functoriality result (like the base change in the abelian case) which pins down this correspondence and also can be checked directly from the construction. This is my own research goal, and I haven't been as successful as I hoped, apart from some special cases, but one thing that I would like to do in this course (so that I don't bore you too much with the flood of new notions which I can't elaborate anyway) is to compare our construction with the abelian case. We can check that our construction actually specializes to what we did in Lecture 1 and 2, when we set n=1. And change of \kappa should shed some light on base change and induction from characters for unramified extensions - we'll see how it goes. But first we have to learn something about the **l-adic etale cohomology** and the **l-adic Galois representations**.

(30/10) I put the barely legible notes from the yesterday's colloquim talk here. I'm putting it on my main webpage too - there are some more scanned notes etc there, if you're looking for something related to Langlands stuff. See you tomorrow.

(29/10-II) Those who came to my colloquim talk - thanks for coming! I felt much better when I saw you guys, as I was a bit nervous in the beginning (as you may have noticed?) to have such a wide-ranging audience. Thanks to your moral support, I enjoyed giving the talk in the end! If you're interested in the division equation of lemniscates, I have incomplete notes here. Nathan asked about the reference for my talk; I refer you again to Taylor's article "Galois representations" in his website. Carl asked about p^{-m}/O in my lecture - my p^{-m} is a fractional ideal in K, i.e. an O-module generated by \varpi^{-m} where \varpi is a uniformizer. Then p^{-m}/O is the quotient O-module, it is a finite set of cardinality q^m, isomorphic to O/p^m (but not canonically! - that's why I'm using p^{-m}/O). In case O=Z_p, it looks like (1/p^m)Z/Z - looks more familiar? See you on Wednesday!

(29/10) I decided not to dwell on completing the proof from last lecture, as we saw the main statements and diving into the computation again wouldn't be so illuminating anyway... and you can check them on the notes. As I mentioned, the toughest chunk in the first half on the course is over - for a while, we'll happily introduce more structures on it and see how rich these objects are. On the other hand, I may introduce more notation and definitions, so remember to stop me whenever the new notation gets confusing. I have to think about the colloquim talk for now!

**Week 3** (22/10/07 - 26/10/07)

(26/10) Whew, finally wrote up the proof for today's theorem - we defined the deformation functor of a formal O-module over a field contained in \bar{F_p}, and showed that this is formally smooth of dimension (height)-1. (The latter part of the notes is still a bit clumsy - well the bulk of p.4 is just the "O-version" of the Lemma 1 of Lecture 5; a p-adic congruence between binomial coefficients (and its extension to O-multiplication) - I'll try to rewrite it when I get more time.) The basic idea of the computation of tangent space is to construct an isomorphism from a given deformation to a deformation with the standard form, and it's very similar to what we did in Lecture 6. Except that the congruence is a bit more refined, as we look at what f(X) congruent to X mod deg q^i will do to Sigma mod deg q^{i+h} - such computation is doable only because we are focusing our attention to the case where coefficient ring is k[epsilon] - and treating this simple case enables us to control deformations to any complete noetherian local ring, by the machinery of Lecture 7. Next week I'll complement on this proof - so take a look at the main Thm at the bottom of p.5. As you see, we have proven the **representability** of our deformation functor by checking that TDef(k) is finite-dimensional. We'll call this deformation space SpfO_k[[T_1,...,T_{h-1}]] the **Lubin-Tate space**, as they considered the case O=Z_p in the 60s. The latter statement about the coordinates hasn't been quite proven in the notes yet, but we're very close - we just have to take a "good" lift of the basis of TDef(k) to Def(O_k[[T_1,...,T_{h-1}]]).

Now some remarks on the stuff "out there". This coordinate system desribes the Newton polygon stratification on the unitary Shimura variety of Harris-Taylor type. When h=2 and O=Z_p, if we combine what we proved with the **Serre-Tate theorem** [stating that the deformation of Elliptic Curves (or more generally Abelian Varieties) is equivalent to the deformation of the associated p-divisible groups (formal group is an example of p-divisible groups)], we get the smoothness / Z_p of the integral model of modular curves when p does not divide the level. So, what we are doing now looks like a weird bulk of power series computations, but it lies at the technical heart of foundational statements of modern arithmetic geometry. There is a vast generalization of this deformation theory to the formal moduli of p-divisible groups (so-called **Rapoport-Zink spaces**), but our basic case (and its dual case, the Drinfeld upper half spaces case) is still the only example where we can do everything over arbitrary base O - usually the ramification of O over Z_p poses technical difficulties (not insurmountable in the future, I suppose), whereas in our case it works uniformly for any O (the ring of integer of our local field K) - that's why it was powerful enough to be used to prove the local Langlands correspondence for GL_n (the global proof doesn't proceed unless we change the base field freely - just as in the proof of global class field theory - although we have very poor understanding of what base change is doing "locally"). Anyway. We're close to the end of the deformation theory part - this more or less corresponds to Lecture 1 in the abelian case where we constructed Lubin-Tate groups (formal O-module of height 1) - we'll proceed to the analogue of Lecture 2 - constructing ramified extensions (coverings) of this smooth deformation space.

(24/10) We had categorical exercise number two - but it's kind of nice to see these abstract reformulation yielding nontrivial results; category theory in action. The two themes we treated today - smooth deformation functor and a functor classifying all formal modules - are dealt with the similar principle, namely reducing polynomial algebra problem to linear algebra problems. To analyze a **Hom set between rings** (which are tantamount to solving a set of polynomial equations), you fibrate it successively so that its fibers are **Hom sets between modules** (i.e. solving linear equation). It's absolutely important to realize that all these categorical languages (that you'll see everywhere in research papers) are actually translation of "equations that we're solving", Hom set being set of solutions, and functors are basically translating one equation to another, or reducing the general solution to a "universal" solution. It was kinda good that we did that for linear equation in Lecture 5. If you're new to this and finding hard to follow, Don't Worry! It is like learning a new language, and simply takes time; it has little to do with how good you are in mathematics in general, in my opinion. I'm going through this rather quickly on purpose - please do check the details on the scanned notes by yourself if you have time. On Friday we'll get back to computations (but very little) of power series.

(22/10) Sorry that I suddenly had to jump into categorical stuff and I hope I didn't scare you too much. I tried to be careful not to use any notion that are not contained in my prerequisite notes (exception; Artinian rings were not defined...), so if you aren't so familiar with the categorical language, especially represented functors, additive categories etc, just look into my Algebraic Number Theory notes. Some people call these stuff "categorical nonsense" - well I understand what they mean, but I think these stuff are fun, no nonsense. But it's very time consuming to make a coherent formulation for yourself when you try to prove what you need. As I couldn't prove my main theorem today I'll cut down my plan for this week; on Wednesday I'll make a brief summary of what we did today and prove the main theorem (formally smooth deformation functor with fin dim tangent functor is pro-represented by a power series ring), and then start the discussion of Lazard's ring, and define the deformation functor of formal O-modules. As I indicated, the deformation functor of a formal O-module of height n over \bar{F_p} will be formally smooth of dimension n-1. I will finish the computation of tangent functor by Friday.

Bao remarked that there was a flaw in the proof of Cor.2 i) of Lecture 6 - at the induction step, I'm applying Cor 1 to f_n Sigma f_n^{-1}, which will have a different [\varpi] from the initial one. I guess I was a bit rash to state it in this generality - so, when we apply this statement to prove the main theorem of Lecture 6, our [\varpi] are both equal to X^{q^h}, and we're dealing with isomorphisms defined over F_{q^h}, so [\varpi] wouldn't change at all, so it works in this particular case! I wondered if similar thing will work in a bit more general setting but I haven't found a suitable generalization, so I'd better restrict my statement of Cor.2 i) to this special case... I've rewritten the notes (check below). On the other hand, we will see that this statement "[\varpi] determines Sigma" will be true for deformations (which might be why I got mixed up).

**Week 2** (15/10/07 - 19/10/07)

(19/10) Classification over \bar{F_p} proven; cheers! A plan of next week: (1) Generalities on smooth (unobstructed) **Deformation Theory**; (2) Lazard's ring and deformation of formal O-modules (**Lubin-Tate Spaces**), (3) Drinfeld level structures and **Lubin-Tate Tower**. As in the Lubin-Tate theory, we are interested in the formal O-modules (of higher height) over some ring of integers of local fields. In the height 1 case, it was unique up to isomorphism over \hat{O}, after all. In the higher height case there will be many of them, actually an (n-1)-dimensional family of such things. Thus, to get a canonical object, we want to know the whole parameter space, or a **universal** formal O-modules, not over CDVR any more, but possibly bigger **complete noetherian local rings**. Any formal O-module over a CDVR will be one of its specializations. So we're thinking of formal O-module over local rings as a **deformation** of one over \bar{F_p}. (When n=1 the deformation was unique and we could directly build the theory over CDVRs.) The deformation space of the height n formal O-module over \bar{F_p} (**Lubin-Tate space**) turns out to be a formal power series ring of n-1 variables over \hat{O}, and we will prove this (again O-Lazard's lemma is essential.) The analogue of the Lubin-Tate extension (they were totally ramified!) will be the ramified coverings (**Lubin-Tate tower**) of Lubin-Tate space, defined as the moduli space of Drinfeld level structures - they will not be formally smooth over \hat{O} any more. The underlying theme of these lectures will be **moduli space** or **classifying space**, which is one of the central notions in modern number theory / algebraic geometry that pops up everywhere. (In the notes below page 3 is not strictly need for next week - I added a comment on the endomorphism ring of a height n formal O-module over \bar{F_p} - it turns out to be the maximal order of the division algebra over K with invariant 1/h - we may come back to that in the future.)

(17/10-II) Some additional comments. Brandon asked about the relation of the local Artin map with the global class field theory. Take a number field F, and consider the local Artin map that we saw for all completion: Art_v : F_v --> Gal(F_v^ab / F_v). The main theorem of the global class field theory is that, if you take the product of this map for all v (including the infinite places where the local Artin map is trivially defined) and define Art_F : A_F^* --> Gal(F^ab/F), where A_F^* is the idele group, then (i) it factors through A_F^*/F^* (idele class group), and (ii) it gives an isomorphism between Gal(F^ab/F) and A_F^*/ (closure of F^*F_\infty^0) where F_\infty^0 is the connected component of (F tensor R)^*. An ideal class group or any ray class group of ideals in F are canonically realized as a quotient of the idele class group. Kaloyan asked about proving [-1]_F\circ F = F\circ [-1]_F for a formal group F. I copy my reply here: "Mimic the proof in the (usual) group theory - how did you prove -(x+y) = (-x) + (-y) for abelian groups? One way is that, You first show that an inverse, if exists, is unique, and check that RHS is indeed the inverse of x+y, using associativity and commutativity. Now [-1] was the unique power series satisfying F( X , [-1](X) ) = 0 so [-1]\circ F is the unique power series satisfying

F( F(X,Y) , [-1]\circ F(X,Y) ) = 0

so we check that F\circ [-1] = F( [-1](X) , [-1](Y) will have this property too.

F( F(X,Y) , F( [-1](X) , [-1](Y) ) )

= F( F(Y,X) , F( [-1](X) , [-1](Y) ) )

= F( Y , F( X, F( [-1](X) , [-1](Y) ) )

= F( Y , F( F(X, [-1](X)) , [-1](Y) ) )

= F( Y , F( 0 , [-1](Y) ) ) = F( Y , [-1](Y) ) = 0

In fact, all the natural identity you'd like to prove for formal
groups can be proven with the same strategy - mimic the proof for
abstract groups. A deeper reason that lies in the background of this
is that the formal groups are actually "formal group schemes", which
are functors from a certain category to the category of sets, and are
"group functors", which means that this functor takes values on
groups, not just sets. So all the properties of this functor can be
recovered from its "valued points". "

(17/10) Sorry I ran a bit late today - I wrote up the complete proof on the uploaded notes. The whole point of today's lecture was that perturbation of formal groups (or formal O-modules) is very limited, and at each degree (of the power series) we can explicitly determine how much perturbation we can have, because it becomes a homogeneous linear equation. And fortunately, a set of homogeneous linear equation over PID (such as Z or O) has a universal solution in that PID - that's what the "structure theorem of finitely generated modules over PID" means - we can write down explicitly what they are. In this case, the answer is that the only perturbation we can have is just a scalar multiple of the most stupid guess (the binomial coefficients B_n), except that when the degree is a power of p you can divide B_n by p. Now this slight modification is crucial - these "bumps" at p-power (or q-power in O-module case) is where you start seeing non-isomorphic formal modules. On Friday we'll start by seeing that as long as the perturbation is a multiple of B_n, they are isomorphic to each other (mod deg n+1).

(15/10) I know I'm still going fast - if I start going through these lemmas on power series slowly, we may feel more comfortable about the proofs, but on the other hand it can get boring, and more importantly, we may lose sight of the structure of the topic. When I prove something on the board, my modest goal is to convince you that they are proven, or at least provable, by the tools we have at hand. (And perhaps that the lecturer feels comfortable with the proof - which can be a rather rare situation in advanced courses, actually!) So relax - it's okay if you can't follow the proofs on the spot, you can review it on my scanned notes if you like.

(14/10) I typed up the whole Lecture 1-3, including the proofs of lemmas that I omitted in the lectures, and added it into the number theory notes (in particular, I added a page on complete unramified extensions in Section 20 - Mahesh asked me about the completed unramified extensions, and then I realized that the fixed field of \phi^n in \hat{K} being K_n isn't completely trivial - do check it out):

I don't think I keep on adding typed stuff into this long notes (for now). Tomorrow I finish up the last week's material by stating the **Local Class Field Theory** (the one main theorem as I see it) and the its relation to the Artin maps obtained by Lubin-Tate theory and Galois Cohomology.

After that we'll move on to the generalities of **Formal Groups** - no sophisticated arithmetic, just algebra - the key lemma of Lazard is something that works over completely general commutative rings. Our first goal is the classification of formal groups and formal O-modules over fields; in fact, we will see that any formal groups over separably closed fields are isomorphic to one of those that we have already seen. Tomorrow I'll start by showing that **all formal groups over a field of char.p are formal Z_p-modules** (actually it's true even over CDVRs with char.p residue field), and then introduction to Lazard's lemma - hopefully prove the lemma on Wednesday and classification on Friday. Next week we recast the lemma into **Lazard's ring** (a moduli space of all formal groups "not up to isomorphism"), to apply it to the deformation problems to mixed characteristic complete local rings.

**Week 1** (08/10/07 - 12/10/07)

(12/10) Cheers for keeping up for the whole week - if you didn't make it this morning, come back on Monday! We finished the proof of Lubin-Tate theory; I left out the proofs of some of the lemmas, but will follow them up in uploaded notes. Today I let myself loose on non-rigorous babbling a bit (certainly I'll do so more in the latter half of the term) - this general principle of converting Galois action into an algebro-geometric (polynomial/power series) action is very important. Next week I'll follow up a little on the logical relation between LT-theory and LCFT (I should state LCFT!), and then start a general theory of formal O-modules (first mainly over F_p or \bar{F_p}, and then deformation to O_L); if you're lost in Galois theory, don't worry, no Galois theory for a while! I incorporated Lecture 1 into the typed notes below.

Bao remarked: i) you can use Eisenstein's irreducibiliy criterion for f_m/f_{m-1} in the Prop.1-(2) of Lecture 2: certainly. ii) in the proof of Claim in Thm 2 of Lecture 3, you are computing in the composite field \hat{K}^m = K^m_x \hat{K^ur}, although the values are in K^m_x: certainly. (But note that all the power series have coefficients in \hat{K^ur}.) Andreas asked if the implication " if N(pi)=N(pi'), then F_f\isom F_{f'} over O_L" of Prop.2 of Lecture 2 has its converse. If you have an isom h : F_f --> F_{f'} such that f'h = h^\phi f, then h has to be of the form [theta] and the Prop shows that N(pi)=N(pi'). Now we'll see next week that the condition f'h = h^\phi f is equivalent to ask for h to be a hom of **formal O-modules** over O_L (not just as formal groups over O_L), and those are the ones we are interested in. We will work systematically with formal O-modules from next week - in particular, any formal O-module of height 1 over O_L turns out to be a Lubin-Tate group (F_f, [ ]_f) for some f in O_L[[X]].

(10/10) I tried to slow down a bit - I'm glad you're asking more questions - please do so more!

(08/10) It was truly great to see so many of you again. I realized my typed pdf file contained some letters asking for Japanese fonts; I just fixed that - hope it's ok now. I put the scan of my scribbled notes here.

(07/10) Okay, I typed up 6 pages of the basics of Local Fields - it is attached to the prerequisites notes (which is also revised) as I refer to many theorems there. Lubin-Tate theory should follow these notes in a next few days. One thing which might look rather strange is that I do not use any topology in these notes - at this level, topological arguments about local fields are easily translated into basic algebra (local rings, inverse limits etc), and the analogies with real/complex fields do not emerge yet. I hope you find these self-contained notes more accessible than say Serre's Local Fields (which refer to Bourbaki at some crucial points).

I still plan to jump into the theory of formal groups over the rings of integers of local fields. Among the basics of local fields, what you need to feel comfortable with is the **Frobenius automorphism** on it, which becomes the q-th power map when reduced mod maximal ideal.

My rough plan for the 3 lectures this week is (1) **Lubin-Tate groups**, (2) **Lubin-Tate extensions**, (3) **Local Class Field Theory**. We do not treat the proof of the base change property nor the local Kronecker-Weber theorem, but will cover the essentials of Lubin-Tate theory. I may incorporate the material of Week 1 into the above notes. Our treatment will keep the higher height case (Week 2) in view - formal modules of arbitrary height n over F_q is obtained by reducing Lubin-Tate groups modulo p. The highlights of Week 2 will be (1) **Lazard's theorem**, (2) **Classification of formal groups over \bar{F_p}**, and will start recasting Lubin-Tate theory into the framework of Non-abelian Lubin-Tate theory.

For those who are interested in the global Langlands stuff that I talked about in the first lecture - make sure you check out R. Taylor's article "Galois Representations" from his website.

**Week 0** (05/10/07)

(05/10) It was a pleasant surprise to see so many of you in the first meeting! I hope you got something out of it, even if you decide not to come next week. The lecture was based on my past talk at Harvard, and I scanned the notes here:

Next week I'll start from the construction of Lubin-Tate groups, just some manipulation of formal power series over CDVR's, no more obscure abstract stuff (for a while). I'm working on the notes on Local Fields continuing the prerequisite notes - I'll put it here when it's done.

(03/10) The first lecture will be an overview of the course.This 100-page note starts "from the scratch" and proves up to the prime decomposition law in cyclotomic fields and quadratic reciprocity. If you flip through this and feel that you are reasonably comfortably with the material in it, you do know enough to attend the course (provided that you are motivated to be exposed to many new stuff!). We'll start by reviewing the basic theory of local fields, which can directly follow this note. To have some flavour of where we start next week, see the first few pages of the following notes:

maintained by Teruyoshi Yoshida

Last modified: Oct 29, 2007.

Last modified: Oct 29, 2007.