# Taniyama-Shimura 4: The Conjecture

We’ve done a lot of work so far just to try to define the terms in the Taniyama-Shimura conjecture, but today we should finally make it. Our last piece of information is to write down what the L-function of a modular form is. Since I don’t want to build a whole bunch of theory needed to define the special class of modular forms we’ll be considering, I’ll just say that we actually need to restrict our definition of “modular form” to “normalized cuspidal Hecke eigenform”. I’ll point out exactly why we need this, but it doesn’t change anything in the conjecture except that every elliptic curve actually corresponds to an even nicer type of modular form.

Let ${f\in S_k(\Gamma_0(N))}$ be a weight ${k}$ cusp form with ${q}$-expansion ${\displaystyle f=\sum_{n=1}^\infty a_n q^n}$. Since this is an analytic function on the disk, we have the tools and theorems of complex analysis at our disposal. We can perform something called the Mellin transform. It is just a standard integral transform given by the formula $\displaystyle {\Lambda (s) = \int_0^\infty f(it)t^s\frac{dt}{t}}$.

After some computation you find that this transformed function is a product of really nice functions. We get $\displaystyle {\Lambda (s)=\frac{N^{s/2}}{(2\pi)^s}\Gamma(s)L(f,s)}$, where ${\Gamma(s)}$ is the usual Gamma function. Now if you actually went through and worked this out you would find out that ${L(f,s)}$ has a really nice form in terms of the Fourier coefficients. The so-called L-series associated to the Mellin transform is given by

$\displaystyle \displaystyle L(f,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$.

If your eyes glazed over for the Mellin transform talk, then just think of the L-function of the modular form as taking all of its Fourier coefficients and throwing them in the numerator of this series to make a new function. A quick remark is that if all the ${a_n}$ are ${1}$ (this won’t happen) we recover the Riemann zeta function. Thus you could think of the L-function we get as some sort of generalization of the zeta function. If you’ve been through some elementary number theory you have probably even seen a proof that $\displaystyle {\sum_{n=1}^\infty \frac{1}{n^s}=\prod \frac{1}{1-p^{-s}}}$ where the product is over all primes called an Euler product. Now in general if I hand you a sequence of integers ${a_n}$ that has some reasonable growth condition, then ${\sum_{n=1}^\infty \frac{a_n}{n^s}}$ will be a nice convergent series, probably with an analytic continuation to the plane. The tricky part is to figure out what types of sequences allow this Euler product decomposition.

This is where we have to use that ${f}$ was of this special form. In the theory of modular forms there is something called Atkin-Lehner theory which tells us that the ${a_n}$ for a cusp form of this special type actually satisfy some nice relations such as ${a_{nm}=a_na_m}$ when ${(m,n)=1}$. These relations are precisely the ones needed to conclude that there is a nice Euler product expansion and it is given by

$\displaystyle \displaystyle L(f,s)=\prod_{p|N}(*)\prod_{p\nmid N} \frac{1}{1-a_pp^{-s}+p^{k-1-2s}}.$

We say that a variety is modular if ${L(X,s)}$ coincides with ${L(f,s)}$ up to finitely many primes for some ${f\in S_k(\Gamma_0(N))}$. We’ve been ignoring the technicalities of dealing with the primes of bad reduction and the primes that divide the level (a surprisingly hard problem to determine when these are the same set!), but now we see that for the definition of a variety being modular this doesn’t even matter. There are other subtleties in defining all of this for when the variety does not have ${2}$-dimensional middle cohomology, but again for our immediate purposes you can trust that people have made the suitable adjustments.

Now we see the truly shocking results of Taniyama-Shimura. We take this incredibly symmetric analytic object (so symmetric it is surprising any exist at all) and we take this completely algebraic variety defined over ${\mathbb{Q}}$ and the conjecture claims that we can always find one of these symmetric things that match up with this action on the cohomology. Wiles and Taylor are often credited with proving it in 1994, but it wasn’t actually proved until 2003 by Breuil, Conrad, Diamond, and Taylor. This was the elliptic curve case.

Just last year Gouvea and Yui proved that all rigid Calabi-Yau threefolds are modular. It is a conjecture that all Calabi-Yau varieties over ${\mathbb{Q}}$ should be modular, so this includes K3 surfaces. It might seem weird that K3 surfaces haven’t been proven but the threefold case has been. This just has to do with those technicalities of what to do if the middle cohomology is bigger than 2-dimensional, which it always is. There you have it. The famous Taniyama-Shimura conjecture which led to a proof of Fermat’s Last Theorem.

# Mirror Symmetry A-branes

I started writing this post this past weekend, but got stuck really quickly and then kept putting it off. I don’t want to leave anyone following this hanging with no idea what the A-model is. This is harder for me to describe than the A-model for some reason. Mostly I’m running into the problem of either just saying what the A-side is without explanation or I’m getting too bogged down in details. Both seem bad. In conclusion, I think I’ll err on the side of too few details, and then hopefully make sense of what is going on by completely describing mirror symmetry in the easiest case possible: the one dimensional case, i.e. for an elliptic curve.

I’m going to semi-cheat right off and refer to posts over a year old. Recall what a symplectic form is on a smooth manifold is. It is just a closed non-degenerate 2-form. A smooth manifold plus symplectic form is called a symplectic manifold. The cotangent bundle always has a canonical symplectic form on it. An example that may be less well-known is that any smooth complex projective variety is symplectic because the Fubini-Study Kähler form on ${\mathbb{P}^n}$ restricts to a symplectic form.

If we just think about vector spaces for a second, then given a symplectic form, we say that a subspace ${S}$ is isotropic if ${S\subset S^\perp}$ and coisotropic if ${S^\perp \subset S}$. The subspace is Lagrangian if it is both isotropic and coisotropic. This extends to manifold language easily by saying an embedded submanifold ${S\subset M}$ is Lagrangian if the tangent subspace ${T_sS\subset T_sM}$ is Lagrangian for every point of ${S}$. If you want to get used to these definitions, a quick exercise would be to check that the zero section of the cotangent bundle is Lagrangian with respect to the canonical symplectic structure.

My second semi-cheat is to ask you to recall the definition of an almost complex structure from close to two years ago. The way to think about it is that it is a bundle map ${J: TM \rightarrow TM}$ that behaves similarly to “multiplication by ${i}$“. The condition is that ${J^2=-Id}$, and indeed multiplication by ${i}$ when identifying ${\mathbb{R}^2\simeq \mathbb{C}}$ gives an example of an almost complex structure. In fact, since we’ll always work over ${\mathbb{C}}$, any complex manifold does have multiplication by ${i}$ as a natural almost complex structure.

It is possible that all these things are related by the following. Suppose ${(M, \omega)}$ is a symplectic manifold, ${J}$ an almost complex structre, and ${g}$ a Riemannian metric. These three structures are called compatible if ${\omega(J(-), -)=\langle - , -\rangle_g}$. I am far out of my depth here, but I’m pretty sure such a manifold is called Kähler if this happens, but maybe some slight more conditions are needed (e.g. does this automatically imply that ${g}$ is Hermitian? If so, then this is definitely what people call Kähler).

Now for the definition of the A-model. Let ${(M, \omega)}$ be a Kähler (in the sense of the previous paragraph) manifold. We define the Fukaya category ${Fuk(M)}$ to have as objects the Lagrangian submanifolds. The morphisms require a bit of technicality to define, but essentially are a way to intersect the submanifolds. It involves all the structures above and is called Floer cohomology. Recall that we’re merely sketching an idea here! Somehow this should be an ${A_\infty}$ or dg-category if you remember from last time, and this just comes from the fact that the morphisms have to do with cohomology classes of intersections.

If you’ve been following this at all, then you should be in utter amazement. We can state mirror symmetry now as an equivalence of ${A_\infty}$ categories ${D^b(X)\rightarrow Fuk(\widehat{X})}$ where ${X}$ is a Calabi-Yau. Why is this amazing (for those not following along)? Look at the left side of this equivalence. The bounded derived category of coherent sheaves (in the Zariski topology!!) on ${X}$ is something that has to do purely with the algebraic data of ${X}$. I mean, the Zariski topology is algebraic, the definition of coherent is very algebraic, the construction of the derived category is algebraic, etc.

The right hand side seems to have forgotten all of the algebraic data. You forget that it is a variety and instead think of it as a smooth manifold. You consider a bunch of structure that helps you study the smooth structure. You consider Lagrangian submanifolds. The Fukaya category is almost entirely analytic in nature. But now the conjecture of Kontsevich mirror symmetry is that the two are always equivalent. That’s it for today. There should be one more post in this series in which I try to sketch the conjecture in the case of an elliptic curve.

# Quick Update

I will be doing the Mirror Symmetry thing. I had someone “like” the post which I’ll count as a vote for it and someone explicitly vote for it. That’s two whole people! For all I know that is half of my regular readers, so by majority rule I have to do it.

I haven’t gotten around to a post because I’ve been incredibly busy. I’ve been doing my usual research and I’m taking two classes this quarter. In addition to all that I’ve been teaching a “mini-course” in our AG club. So for the next several weeks when I have down time I’ll probably think about what I’m going to say there and/or I’ll be typing up notes for that. I should point out that mirror symmetry has two parts, the derived category side and the Fukaya category side. The derived side is something that is really close to what I do, so it is without a doubt worthwhile to blog about that half, and I plan to soon. I may be a bit sketchier on the side that isn’t as important to me.

I’ll just leave you with a quick taste of what some people mean by “Mirror Symmetry” which is a bit different than Kontsevitch Mirror Symmetry which is what I’ll be explaining at some point. Suppose you have a Calabi-Yau threefold. By this I’ll mean a smooth, projective variety of dimension three over $\mathbb{C}$ with $\omega_X\simeq \mathcal{O}_X$ and $H^1(X,\mathcal{O}_X)=H^2(X,\mathcal{O}_X)=0$. By general Hodge theory there is a symmetry in the Hodge numbers $h^{pq}(X)=\dim_{\mathbb{C}}H^q(X, \Omega^p)$, namely that $h^{pq}(X)=h^{qp}(X)$. Also, you can use the fact that it is Calabi-Yau to check that all Hodge numbers are completely determined (independently of $X$) as either 0 or 1 except $h^{11}$ and $h^{12}$. Fun exercise in Serre duality!

Suppose $X$ is a Calabi-Yau threefold. There is some specified $h^{11}$ and $h^{12}$ (the only two unknown Hodge numbers). A mirror pair for $X$ is another Calabi-Yau threefold with $h^{11}$ and $h^{12}$ swapped. In my brief encounter with Kontsevich Mirror Symmetry (which says something about an equivalence of categories) this will follow as a special case.

Since I’m in the mood I may as well say some things that immediately pop into mind when seeing this as someone that has recently been thinking in the arithmetic world. If we are over an algebraically closed field of characteristic 0, then there is a result that says $h^{11}>0$. In particular, there cannot be a rigid CY 3-fold if mirror symmetry is true, since $h^{12}$ gives the space of deformations of $X$. But in positive characteristic there are tons of rigid CY 3-folds! Interesting.

I’ll leave you with that little taste of what is to come.

# Mirror Symmetry

Well, I keep putting off writing a new post because I’m not sure what I’m going to do it on. I had an idea. I work with Calabi-Yau varieties a lot, and so inevitably the term “mirror symmetry” appears all over the place. I’m mostly interested in arithmetic properties of Calabi-Yau’s where mirror symmetry doesn’t apply, so I know absolutely nothing about it. Since I’m curious what is meant when people use this term I thought I’d do a series of posts trying to explain the main idea of mirror symmetry.

In fact, a week or so ago Matt Ballard (who graduated the year I started grad school at the same school) put up on the archive a really nice introduction to the subject. This means I even have a nice reference to work with now. Here’s the problem. To type up something fairly reasonable on the subject is going to be a major undertaking. I have very little old blog material that is relevant, so I’ll basically be starting from scratch. It is also going to be quite time consuming since I know nothing about it.

This is my dilemma. I’d love to learn about it and blog about it, but I’m not sure it is worth the time and effort it will take. I’m at that point of grad school where I probably shouldn’t go off on a wild tangent for a long time just because I want to know what this term means when it has basically nothing to do with my research. On the other hand, one of the purposes of this blog is to keep me doing things that aren’t directly related to my research so that I maintain some sort of breadth.

I’m just throwing the idea out there for you all. What do you think? Do you have an interest in hearing about (homological) mirror symmetry, because if there seems to be interest, then it is probably worth doing.

# Bogomolov-Tian-Todorov Theorem

I came up with this interesting idea of blogging the proof of what is commonly called the Bogomolov-Tian-Todorov theorem. It will probably be only one post as I lightly sketch the proof. Originally I was going to go into the details, but I’ve spent several hours going through it now, and it is mostly just tedious calculations.

What does the theorem say? In its current generality it says that if ${X}$ is a Calabi-Yau variety (actually a little weaker than that even) over an algebraically closed field of characteristic ${0}$, then the deformation functor is formally smooth. As a corollary we get that deformations of Calabi-Yau’s are unobstructed. This theorem and proof have quite a tangled history.

The three names attached to it correspond to a complex manifold proof. It only works over ${\mathbb{C}}$ and uses all sorts of analytic stuff including complex structures. It is kind of horrifying to an algebraic geometer and has no hope of generalizing to Calabi-Yau varieties over other fields (including char ${0}$ alg closed ones). Then Ran in 1992 came up with a slightly more algebraic method for proving it over ${\mathbb{C}}$ with jet bundles. Kawamata extrapolated a slick purely algebraic way of proving it from that. It over other fields of characteristic ${0}$ and then Fantechi and Manetti cleaned up a few things from Kawamata and removed a few hypotheses.

The Kawamata-Fantechi-Manetti proof is the one I’m going to sketch for you. Lastly, I’d like to point out a really interesting alternative approach by Iacono and Manetti which uses the idea of ${L_\infty}$ structure and dg Lie algebras to give a very algebraic proof. At some point I’d like to understand that way of doing it. Lurie gave an ICM address about how deformation problems are somehow equivalent to the data of ${L_\infty}$ algebras or something to that effect which might give some partial extensions to positive characteristic (I really don’t know as I haven’t looked into this at all).

As I’ve pointed out, this proof had very concrete geometric origins, but successive generalizations have altered it into basically abstract algebraic formalism. If you look at the mentioned papers in order it won’t seem that weird, but this approach is definitely the most streamlined and clean with the best results. Since I’m not motivating it much I’ve decided to omit the details.

The proof comes in basically two steps. The first purely formal theorem is sometimes called the Kawamata ${T^1}$ lifting theorem. We’ll fix an algebraically closed field, ${k}$, of characteristic ${0}$. Let ${Art_k}$ be the category of Artinian local algebras over ${k}$ with residue field ${k}$. We will call a functor ${F: Art_k\rightarrow Set}$ a deformation functor if it admits a hull and an obstruction theory. The terminology deformation functor suggests that whenever our functor is ${Def_X}$ for some reasonable variety ${X/k}$ it will satisfy these two conditions.

For those who have not seen this before a hull is a prorepresentable functor ${G}$ that maps smoothly ${G\rightarrow F}$. A hull should be thought of as some sort of weak form of a representing object. If ${F}$ is prorepresentable, then the prorepresenting object is a hull, but in general if it is not prorepresentable, then there could be lots of non-isomorphic hulls. The theorem says that if ${F}$ is a deformation functor that satisfies the ${T^1}$-lifting condition, then ${F}$ is smooth.

So what is the ${T^1}$-lifting property? It is the condition that the natural map ${F(k[x,y]/(x^{n+2}, y^2))\rightarrow F(k[x,y]/(x^{n+1},y^2))\times_{F(k[t]/(t^{n+1}))} F(k[t]/(t^{n+2}))}$ is surjective for all natural numbers ${n}$. A more illuminating way of saying this is the following. If ${A_n=k[t]/(t^{n+1})}$ and ${B_n=k[x,y]/(x^{n+1}, y^2)}$, and ${\beta_n: B_n\rightarrow A_n}$ sends ${x\mapsto t}$ and ${y\mapsto 0}$, then for some object ${X_n\in F(A_n)}$ (an ${n}$-th order deformation of ${X}$ maybe), form the set ${T^1(X_n/A_n)=\{Y_n\in F(B_n): F(\beta_n)(Y_n)=X_n\}}$. These are the first-order infinitesimal deformations of ${X_n}$ (thinking of smooth varieties) i.e. if ${F=Def_X}$ and ${X_n}$ is a lift to ${A_n}$, then ${T^1(X_n/A_n)\simeq H^1(X_n, T_{X_n/A_n})}$.

The ${T^1}$-lifting property in terms of ${T^1}$ is just that for any ${X_{n+1}\in F(A_{n+1})}$ we have ${T^1(X_{n+1}/A_{n+1})\rightarrow T^1(X_n/A_n)}$ is surjective. Or unwinding a little further, if we can lift ${X}$ to ${A_n}$, then we can lift it to ${A_{n+1}}$. You might object at this point that the ${T^1}$-lifting property stated in these terms is exactly the statement that ${F}$ is unobstructed, but this isn’t true because remember that ${A_n=k[t]/(t^n)}$. This is a very restricted class of lifting that we have to check and Kawamata’s theorem says this suffices to get unobstructedness for deforming over all local Artin rings.

One idea of the proof is to use a well-known fact that a ${k}$-algebra ${R}$ is smooth if and only if every map ${R\rightarrow A_n}$ lifts to a map ${R\rightarrow A_{n+1}}$. Another aspect of the proof is to use the fact that we had to assume an obstruction theory exists for ${F}$. These two ideas play off eachother, and the rest is basically tedious calculations. You could probably almost piece it together yourself from these hints.

Warning, this really, really only works in characteristic ${0}$. It isn’t just that there is no known proof, but there are very simple counterexamples. Suppose ${k}$ has characteristic ${p>0}$. You can check that ${F(A)=Hom_{k-alg}(k[t]/(t^p), A)}$ is a deformation functor that satisfies the ${T^1}$-lifting property (pretty easy exercise to familiarize yourself with the definitions!), but by construction it is representable by ${k[t]/(t^p)}$ and this is NOT smooth. (There is a nice divided power/crystalline work around to this by Schroer, though…).

Alright. So now we know that in order to check that the deformations of a Calabi-Yau are unobstructed we need only check that ${Def_X}$ is actually a deformation functor, and that it satisfies the ${T^1}$-lifting property. Again, this isn’t so bad. For instance, ${Def_X}$ is certainly a deformation functor (almost by creation of the term “deformation functor”). The two Calabi-Yau properties that get used are the fact that ${\omega_X\simeq \mathcal{O}_X}$ and that ${H^0(X, T_X)=0}$ (again, that second fact isn’t necessarily true in positive characteristic since it requires use of Hodge symmetry). This finishes the proof of the Bogomolov-Tian-Todorov theorem, namely that Calabi-Yau’s are unobstructed in characteristic ${0}$.

# Sheaf of Witt Vectors 2

Recall last time we talked about how we can form the sheaf of Witt vectors over a variety ${X}$ that is defined over an algebraically closed field ${k}$ of characteristic ${p}$. The sections of the structure sheaf form rings and we can take ${W_n}$ of those rings. The functoriality of ${W_n}$ gives us that this is a sheaf that we denote ${\mathcal{W}_n}$. For today we’ll be define ${\Lambda}$ to be ${W(k)}$.

Recall that we also noted that ${H^q(X, \mathcal{W}_n)}$ makes sense and is a ${\Lambda}$-module annihilated by ${p^n\Lambda}$ (recall that we noted that Frobenius followed by the shift operator is the same as multiplying by ${p}$, and since Frobenius is surjective, multiplying by ${p}$ is just replacing the first entry by ${0}$ and shifting, so multiplying by ${p^n}$ is the same as shifting over ${n}$ entries and putting ${0}$‘s in, since the action is component-wise, ${p^n\Lambda}$ is just multiplying by ${0}$ everywhere and hence annihilates the module).

In fact, all of our old operators ${F}$, ${V}$, and ${R}$ still act on ${H^q(X, \mathcal{W}_n)}$. They are easily seen to satisfy the formulas ${F(\lambda w)=F(\lambda)F(w)}$, ${V(\lambda w)=F^{-1}(\lambda)V(w)}$, and ${R(\lambda w)=\lambda R(w)}$ for ${\lambda\in \Lambda}$. Just by using basic cohomological facts we can get a bunch of standard properties of ${H^q(X, \mathcal{W}_n)}$. We won’t write them all down, but the two most interesting (of the very basic) ones are that if ${X}$ is projective then ${H^q(X, \mathcal{W}_n)}$ is a finite ${\Lambda}$-module, and from the short exact sequence we looked at last time ${0\rightarrow \mathcal{O}_X\rightarrow \mathcal{W}_n \rightarrow \mathcal{W}_{n-1}\rightarrow 0}$, we can take the long exact sequence associated to it to get ${\cdots \rightarrow H^q(X, \mathcal{O}_X)\rightarrow H^q(X, \mathcal{W}_n)\rightarrow H^q(X, \mathcal{W}_{n-1})\rightarrow \cdots}$

If you’re like me, you might be interested in studying Calabi-Yau manifolds in positive characteristic. If you’re not like me, then you might just be interested in positive characteristic K3 surfaces, either way these cohomology groups give some very good information as we’ll see later, and for a Calabi-Yau’s (including K3’s) we have ${H^i(X, \mathcal{O}_X)=0}$ for ${i=1, \ldots , n-1}$ where ${n}$ is the dimension of ${X}$. Using this long exact sequence, we can extrapolate that for Calabi-Yau’s we get ${H^i(X, \mathcal{W}_n)=0}$ for all ${n>0}$ and ${i=1, \ldots, n-1}$. In particular, we get that ${H^1(X, \mathcal{W})=0}$ for ${X}$ a K3 surface where we just define ${H^q(X, \mathcal{W})=\lim H^q(X, \mathcal{W}_n)}$ in the usual way.