## Serre-Tate Theory 2

I guess this will be the last post on this topic. I’ll explain a tiny bit about what goes into the proof of this theorem and then why anyone would care that such canonical lifts exist. On the first point, there are tons of details that go into the proof. For example, Nick Katz’s article, Serre-Tate Local Moduli, is 65 pages. It is quite good if you want to learn more about this. Also, Messing’s book The Crystals Associated to Barsotti-Tate Groups is essentially building the machinery for this proof which is then knocked off in an appendix. So this isn’t quick or easy by any means.

On the other hand, I think the idea of the proof is fairly straightforward. Let’s briefly recall last time. The situation is that we have an ordinary elliptic curve ${E_0/k}$ over an algebraically closed field of characteristic ${p>2}$. We want to understand ${Def_{E_0}}$, but in particular whether or not there is some distinguished lift to characteristic ${0}$ (this will be an element of ${Def_{E_0}(W(k))}$.

To make the problem more manageable we consider the ${p}$-divisible group ${E_0[p^\infty]}$ attached to ${E_0}$. In the ordinary case this is the enlarged formal Picard group. It is of height ${2}$ whose connected component is ${\widehat{Pic}_{E_0}\simeq\mu_{p^\infty}}$. There is a natural map ${Def_{E_0}\rightarrow Def_{E_0[p^\infty]}}$ just by mapping ${E/R \mapsto E[p^\infty]}$. Last time we said the main theorem was that this map is an isomorphism. To tie this back to the flat topology stuff, ${E_0[p^\infty]}$ is the group representing the functor ${A\mapsto H^1_{fl}(E_0\otimes A, \mu_{p^\infty})}$.

The first step in proving the main theorem is to note two things. In the (split) connected-etale sequence

$\displaystyle 0\rightarrow \mu_{p^\infty}\rightarrow E_0[p^\infty]\rightarrow \mathbb{Q}_p/\mathbb{Z}_p\rightarrow 0$

we have that ${\mu_{p^\infty}}$ is height one and hence rigid. We have that ${\mathbb{Q}_p/\mathbb{Z}_p}$ is etale and hence rigid. Thus given any deformation ${G/R}$ of ${E_0[p^\infty]}$ we can take the connected-etale sequence of this and see that ${G^0}$ is the unique deformation of ${\mu_{p^\infty}}$ over ${R}$ and ${G^{et}=\mathbb{Q}_p/\mathbb{Z}_p}$. Thus the deformation functor can be redescribed in terms of extension classes of two rigid groups ${R\mapsto Ext_R^1(\mathbb{Q}_p/\mathbb{Z}_p, \mu_{p^\infty})}$.

Now we see what the canonical lift is. Supposing our isomorphism of deformation functors, it is the lift that corresponds to the split and hence trivial extension class. So how do we actually check that this is an isomorphism? Like I said, it is kind of long and tedious. Roughly speaking you note that both deformation functors are prorepresentable by formally smooth objects of the same dimension. So we need to check that the differential is an isomorphism on tangent spaces.

Here’s where some cleverness happens. You rewrite the differential as a composition of a whole bunch of maps that you know are isomorphisms. In particular, it is the following string of maps: The Kodaira-Spencer map ${T\stackrel{\sim}{\rightarrow} H^1(E_0, \mathcal{T})}$ followed by Serre duality (recall the canonical is trivial on an elliptic curve) ${H^1(E_0, \mathcal{T})\stackrel{\sim}{\rightarrow} Hom_k(H^1(E_0, \Omega^1), H^1(E_0, \mathcal{O}_{E_0}))}$. The hardest one was briefly mentioned a few posts ago and is the dlog map which gives an isomorphism ${H^2_{fl}(E_0, \mu_{p^\infty})\stackrel{\sim}{\rightarrow} H^1(E_0, \Omega^1)}$.

Now noting that ${H^2_{fl}(E_0, \mu_{p^\infty})=\mathbb{Q}_p/\mathbb{Z}_p}$ and that ${T_0\mu_{p^\infty}\simeq H^1(E_0, \mathcal{O}_{E_0})}$ gives us enough compositions and isomorphisms that we get from the tangent space of the versal deformation of ${E_0}$ to the tangent space of the versal deformation of ${E_0[p^\infty]}$. As you might guess, it is a pain to actually check that this is the differential of the natural map (and in fact involves further decomposing those maps into yet other ones). It turns out to be the case and hence ${Def_{E_0}\rightarrow Def_{E_0[p^\infty]}}$ is an isomorphism and the canonical lift corresponds to the trivial extension.

But why should we care? It turns out the geometry of the canonical lift is very special. This may not be that impressive for elliptic curves, but this theory all goes through for any ordinary abelian variety or K3 surface where it is much more interesting. It turns out that you can choose a nice set of coordinates (“canonical coordinates”) on the base of the versal deformation and a basis of the de Rham cohomology of the family that is adapted to the Hodge filtration such that in these coordinates the Gauss-Manin connection has an explicit and nice form.

Also, the canonical lift admits a lift of the Frobenius which is also nice and compatible with how it acts on the above chosen basis on the de Rham cohomology. These coordinates are what give the base of the versal deformation the structure of a formal torus (product of ${\widehat{\mathbb{G}_m}}$‘s). One can then exploit all this nice structure to prove large open problems like the Tate conjecture in the special cases of the class of varieties that have these canonical lifts.

## Serre-Tate Theory 1

Today we’ll try to answer the question: What is Serre-Tate theory? It’s been a few years, but if you’re not comfortable with formal groups and ${p}$-divisible groups, I did a series of something like 10 posts on this topic back here: formal groups, p-divisible groups, and deforming p-divisible groups.

The idea is the following. Suppose you have an elliptic curve ${E/k}$ where ${k}$ is a perfect field of characteristic ${p>2}$. In most first courses on elliptic curves you learn how to attach a formal group to ${E}$ (chapter IV of Silverman). It is suggestively notated ${\widehat{E}}$, because if you unwind what is going on you are just completing the elliptic curve (as a group scheme) at the identity.

Since an elliptic curve is isomorphic to it’s Jacobian ${Pic_E^0}$ there is a conflation that happens. In general, if you have a variety ${X/k}$ you can make the same formal group by completing this group scheme and it is called the formal Picard group of ${X}$. Although, in general you’ll want to do this with the Brauer group or higher analogues to guarantee existence and smoothness. Then you prove a remarkable fact that the elliptic curve is ordinary if and only if the formal group has height ${1}$. In particular, since the ${p}$-divisible group is connected and ${1}$-dimensional it must be isomorphic to ${\mu_{p^\infty}}$.

It might seem silly to think in these terms, but there is another “enlarged” ${p}$-divisible group attached to ${E}$ which always has height ${2}$. This is the ${p}$-divisible group you get by taking the inductive limit of the finite group schemes that are the kernel of multiplication by ${p^n}$. It is important to note that these are non-trivial group schemes even if they are “geometrically trivial” (and is the reason I didn’t just call it the “${p^n}$-torsion”). We’ll denote this in the usual way by ${E[p^\infty]}$.

I don’t really know anyone that studies elliptic curves that phrases it this way, but since this theory must be generalized in a certain way to work for other varieties like K3 surfaces I’ll point out why this should be thought of as an enlarged ${p}$-divisible group. It is another standard fact that ${E}$ is ordinary if and only if ${E[p^\infty]\simeq \mu_{p^\infty}\oplus \mathbb{Q}_p/\mathbb{Z}_p}$. In fact, you can just read off the connected-etale decomposition:

$\displaystyle 0\rightarrow \mu_{p^\infty}\rightarrow E[p^\infty] \rightarrow \mathbb{Q}_p/\mathbb{Z}_p\rightarrow 0$

We already noted that ${\widehat{E}\simeq \mu_{p^\infty}}$, so the ${p}$-divisible group ${E[p^\infty]}$ is a ${1}$-dimensional, height ${2}$ formal group whose connected component is the first one we talked about, i.e. ${E[p^\infty]}$ is an enlargement of ${\widehat{E}}$. For a general variety, this enlarged formal group can be defined, but it is a highly technical construction and would take a lot of work to check that it even exists and satisfies this property. Anyway, this enlarged group is the one we need to work with otherwise our deformation space will be too small to make the theory work.

Here’s what Serre-Tate theory is all about. If you take a deformation of your elliptic curve ${E}$ say to ${E'}$, then it turns out that ${E'[p^\infty]}$ is a deformation of the ${p}$-divisible group ${E[p^\infty]}$. Thus we have a natural map ${\gamma: Def_E \rightarrow Def_{E[p^\infty]}}$. The point of the theory is that it turns out that this map is an isomorphism (I’m still assuming ${E}$ is ordinary here). This is great news, because the deformation theory of ${p}$-divisible groups is well-understood. We know that the versal deformation of ${E[p^\infty]}$ is just ${Spf(W[[t]])}$. The deformation problem is unobstructed and everything lives in a ${1}$-dimensional family.

Of course, let’s not be silly. I’m pointing all this out because of the way in which it generalizes. We already knew this was true for elliptic curves because for any smooth, projective curve the deformations are unobstructed since the obstruction lives in ${H^2}$. Moreover, the dimension of the space of deformations is given by the dimension of ${H^1(E, \mathcal{T})}$. But for an elliptic curve ${\mathcal{T}\simeq \mathcal{O}_X}$, so by Serre duality this is one-dimensional.

On the other hand, we do get some actual information from the Serre-Tate theory isomorphism because ${Def_{E[p^\infty]}}$ carries a natural group structure. Thus an ordinary elliptic curve has a “canonical lift” to characteristic ${0}$ which comes from the deformation corresponding to the identity.

## Moduli of Vector Bundles on Elliptic Curves

We’ve been talking about moduli problems, and one notoriously hard type of moduli problem is to “classify” vector bundles on some variety. Even when you restrict yourself to some special case like a specific surface and try to classify only vector bundles of certain rank or Chern class you run into trouble. To this day, these types of problems are a very active area of research.

It is fairly well-known (due to Grothendieck, but a problem in Hartshorne as well) that any finite rank vector bundle over ${\mathbb{P}^1}$ is just a finite direct sum ${\oplus_i\mathcal{O}(n_i)}$. The next interesting case would be to move up to genus ${1}$ curves. It turns out that Atiyah in the 1957 paper, Vector Bundles over an Elliptic Curve, worked out a classification. I just learned about this a month or two ago, and it is pretty cool so I’d like to briefly describe the idea.

Fix an algebraically closed field ${k}$ of characteristic ${0}$, and let ${E/k}$ be an elliptic curve. Define ${V(r,d)}$ to be the set of indecomposable vector bundles (up to isomorphism) on ${E}$ of rank ${r}$ and degree ${d}$, where degree just means the degree of the determinant. It is well known that ${V(1,0)}$ can be identified with ${E}$, because ${V(1,0)=Pic^0(E)}$ is the set of degree ${0}$ divisors. In particular, this says that the moduli space of degree ${0}$ divisors (line bundles) on ${E}$ is fine and representable by ${E}$.

Let’s continue with this idea of classifying degree ${0}$ vector bundles. It turns out there is a unique vector bundle ${T_r\in V(r,0)}$ with the property that ${\Gamma (E, T_r)\neq 0}$, i.e. there are non-trivial global sections. Now, recall how ${V(1,0)}$ works. Let ${0\in E}$ be the origin. Then our isomorphism ${E\rightarrow V(1,0)}$ is given by ${P\mapsto \mathcal{O}(-P)\otimes \mathcal{O}([0])}$. Our ${T_r}$ is going to play the role of ${\mathcal{O}([0])}$ here. In complete analogy we get a bijection ${V(1,0)\rightarrow V(r,0)}$ by ${L\mapsto L\otimes T_r}$.

This finishes off the case of degree ${0}$ vector bundles, because we get that ${E\simeq V(1,0)\simeq V(r,0)}$ (roughly speaking, of course there’s a lot more structure we need to know about to say something about the moduli spaces).

In more general situations we can use the same sort of trick. Note that we always have a bijection ${V(r,d)\rightarrow V(r, d+nr)}$ given by ${V\mapsto V\otimes \mathcal{O}(n[0])}$. Thus one reduction Atiyah makes right away is that (by the Euclidean algorithm) we only need to consider ${V(r,d)}$ for ${0\leq d < r}$.

Now suppose the rank and degree are non-zero, and ${n=\text{gcd}(r,d)}$. We can establish a bijection ${E\rightarrow V(r,d)}$, so that the composed map ${E\rightarrow V(r,d)\stackrel{det}{\rightarrow} V(1, d)\rightarrow E}$ is the map multiplication by ${n}$. The idea again is that one can find a certain vector bundle ${T_{r,d}\in V(r,d)}$ that is unique up to isomorphism from which all the others can be produced. Most people call this the Atiyah bundle. As a corollary, we see that if ${(r,d)=1}$, then the moduli space of indecomposable rank ${r}$ and degree ${d}$ vector bundles on ${E}$ is fine and again representable by ${E}$.

In modern language, we could work out that stable or semi-stable sheaves are the appropriate things to classify if we want a hope for the moduli space to be fine. It turns out that this condition of certain numerical invariants being coprime often implies that the sheaves are stable. For example, on an elliptic curve, a K3 surface, or even when dealing with relative moduli of sheaves on a K3 fibration. We may get to this another day.

## L-series of CM Elliptic Curves

This will be the last post in the CM elliptic curve series. Last time we covered the main theorem of complex multiplication. Today we’ll very, very briefly sketch one amazing use of the main theorem. We’ll first talk about how to associate a Grössencharacter to a CM elliptic curve and then use this to better describe the ${L}$-series of an elliptic curve.

Here’s why this will be amazing. Awhile ago we talked about ${L}$-series of varieties and various modularity conjectures. One of the huge, major theorems of modern number theory (which wasn’t proved in full until 2003, and built on all of Wiles and Taylor’s results) is the so-called Modularity Theorem. It says that an elliptic curve ${E/\mathbb{Q}}$ is modular and hence its ${L}$-series has an analytic continuation to all of ${\mathbb{C}}$.

This is still open (as far as I know) for elliptic curves over an arbitrary number field, but today we’ll see that we can use the theory we’ve built to show that any CM elliptic curve over any number field has an ${L}$-series that analytically continues to the whole plane.

Fix ${E/L}$ an elliptic curve over a number field with CM by ${R_K}$, the ring of integers in a quadratic imaginary field ${K}$. We use the main theorem of CM to do the following. Fix an idele ${x\in \mathcal{J}_L}$ and let ${s=N_{L/K}(x)\in\mathcal{J}_L}$. There is a unque ${\alpha\in K^*}$ such that ${\alpha R_K=(s)}$ and for any fractional ${\frak{a}}$ in ${K}$ and any analytic iso ${f: \mathbb{C}/\frak{a}\stackrel{\sim}{\rightarrow} E(\mathbb{C})}$ we get a commutative diagram:

$\displaystyle \begin{matrix} K/\frak{a} & \rightarrow & K/\frak{a} \\ \downarrow & & \downarrow \\ E(L^{ab}) & \rightarrow & E(L^{ab}) \end{matrix}$

What this gives us is a map ${\alpha_{E/L}: \mathcal{J}_L\rightarrow K^*\subset\mathbb{C}^*}$. Recall that a Grössencharacter of a number field ${L}$ is such a map that is trivial on ${L^*}$. We can alter this to the map ${\Psi_{E/L}:\mathcal{J}_L\rightarrow \mathbb{C}^*}$ by ${\Psi_{E/L}(x)=\alpha_{E/L}(x)(N_{L/K}(x^{-1}))_\infty}$. It turns out this is our desired Grössencharacter.

Recall how we formed the L-series of a variety over ${\mathbb{Q}}$. There is nothing special about ${\mathbb{Q}}$ going on in that constuction and so the same thing can be done for any elliptic curve ${E/L}$ where ${L}$ is a number field. Basically you piece it together as a product over primes of some expression involving the characteristic polynomial of the Frobenius elements acting on the cohomology of the reductions of ${E}$ mod these primes.

Given a Grössencharacter ${\Psi: \mathcal{J}_L\rightarrow \mathbb{C}^*}$ we can define the Hecke ${L}$-series to be ${\displaystyle L(s,\Psi)=\prod_{\frak{p}}(1-\Psi(\frak{p})q_\frak{p}^{-s})^{-1}}$, where ${q_\frak{p}}$ is the size of the residue field at ${\frak{p}}$. Hecke proved that this ${L}$-series has an analytic continuation to the complex plane.

Duering proved that if ${E/L}$ is an elliptic curve with CM by ${R_K}$, then ${L(E/L, s)=L(s, \Psi_{E/L})L(s, \overline{\Psi_{E/L}})}$. In fact, even better is that if ${K}$ is not contained in ${L}$, then the ${L}$-series of the elliptic curve over ${L}$ is precisely the Hecke ${L}$-series of the Grössencharacter attached to the base-changed elliptic curve ${E/KL}$. In either case, we see that it is much easier to prove that the ${L}$-series of an elliptic curve with CM by the ring of integers in a quadratic imaginary field has an analytic continuation.

## An Application to Elliptic Curves

Let’s do an application of our theorems about finitely generated projective modules over Dedekind domains. This is another one of those things that seems to be quite well known to experts, but it is not written anywhere that I know of. Suppose ${E}$ and ${F}$ are elliptic curves defined over a number field ${K}$ (this works in more generality, but this assumption will allow us to not break into lots of weird cases), and assume that the ${\ell}$-adic Tate modules are isomorphic for all ${\ell}$.

Recall briefly that the ${\ell}$-adic Tate module is just the limit over all the ${\ell^n}$ torsion points, i.e. ${\displaystyle T_\ell(E)=\lim_{\longleftarrow} E(\overline{K})[\ell^n]}$ as a ${G=Gal(\overline{K}/K)}$-module. We discussed this before in this post. An isogeny ${\phi: E\rightarrow F}$ defined over ${K}$ induces an action via pushforward ${T_\ell (E)\rightarrow T_\ell (F)}$ which is Galois equivariant. In fact, if ${\ell \nmid \text{deg}(\phi)}$, then it induces an isomorphism of Tate modules.

First define ${Hom(E,F)}$ to be the set of isogenies over ${K}$ (this could get me in trouble and has been the main delay in this post). If ${E}$ and ${F}$ are isogenous, then the natural action of ${End(E)}$ by composing turns ${Hom(E,F)}$ into a rank ${1}$ projective module over ${End(E)}$.

The question we want to ask ourselves is how much information do we get from the Tate module. It seems that surely this would not be enough information to recover the curve up to isomorphism, but recall that most elliptic curves do not have complex multiplication. Let’s start with that case. Suppose ${E}$ is non-CM so that ${End(E)\simeq \mathbb{Z}}$. The only endomorphisms are the isogenies given by multiplication by an integer.

The Tate conjecture formally says that there is an isomorphism ${Hom(E,F)\otimes_\mathbb{Z} \mathbb{Z}_\ell \stackrel{\sim}{\rightarrow}Hom_G(T_\ell(E), T_\ell(F))}$ (proved by Faltings in this case). This tells us that if you have some isomorphism ${T_\ell(E)\simeq T_\ell(F)}$, then there is an isogeny that induces it (maybe there is a less powerful tool to see this in this case). But we’ve now assumed that ${End(E)}$ is ${\mathbb{Z}}$, so ${Hom(E,F)}$ is not just a locally free module of rank ${1}$, but just plain free of rank ${1}$. All other isogenies are just composing this one with multiplication by an integer.

Let ${\phi}$ be the generator of ${Hom(E,F)}$. If ${deg(\phi)}$ is ${n}$, then all isogenies are divisible by ${n}$. Since we assume the Tate modules are isomorphic for all ${\ell}$, just pick some ${\ell}$ that divides ${n}$. Since Tate says there is an isogeny inducing the isomorphism we get a contradiction unless ${n=1}$. Thus ${deg(\phi)=1}$ and hence the generator is actually an isomorphism. This proves a fact I’ve seen stated, but haven’t seen written anywhere. If ${E}$ and ${F}$ are non-CM elliptic curves with isomorphic Tate modules for all ${\ell}$, then they must be isomorphic.

This should seem a little strange, because it basically says we can recover the curve up to isomorphism merely from knowing ${H_1}$. It turns out that weirder things can happen for CM curves, but we can use our structure theory from the last post to figure out what is going on. Suppose now that ${End(E)}$ is the full ring of integers in a quadratic imaginary field (the only other possibility is that it is merely an order in such a field).

It turns out that if ${E}$ and ${F}$ have isomorphic Tate modules for all ${\ell}$, then we can’t just conclude they are isomorphic. Here is a good way to think about this. We have that ${End(E)}$ is a Dedekind domain, and ${Hom(E,F)}$ is a rank ${1}$ projective module over it, so it is either generated by ${1}$ element and hence free in which case the same type of argument will show ${E}$ and ${F}$ must be isomorphic. The reason we get no information in the case where it is generated by two things is that these degrees can be coprime. In fact, they must be or else the same argument gives an isomorphism again.

This recently came up in something I was working on, and I couldn’t believe that I couldn’t find this fact stated anywhere (but several number theorists confirmed that this was something they knew). It might be because introductory books don’t want to assume the Tate conjecture, and anything that does assume the Tate conjecture assumes you can figure this out for yourself.

## Geometric Construction of Modular Forms

I was going to move on to deformations of Galois representations to try to talk about some of the ideas involved in proving Taniyama-Shimura, but after a random conversation I decided it might be more fun to try to talk a little about topological modular forms. I know nothing about these things, but it is a pretty big current research interest right now and they are somehow related to the modular forms I just talked about, so let’s give it a try.

This post is filling in more information about modular forms. Rather than make you look up this definition again, I’ll copy it back here and give some motivation for it. A modular form of weight ${k}$ and level ${N}$ is an element of the vector space ${M_k(\Gamma_0(N))}$ which consists of holomorphic functions on the upper half plane ${f:\mathcal{H}\rightarrow \mathbf{C}}$ satisfying the additional transformation property that

$\displaystyle \displaystyle f \left(\frac{az+b}{cz+d}\right)=(cz+d)^kf(z)$

for all matrices ${\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)\in SL_2(\mathbf{Z})}$ such that ${c\equiv 0 \mod N}$ (plus the “cusp” condition). The ones that vanish at the cusp are called cusp forms and they form a vector space denoted ${S_k(\Gamma_0(N))}$.

Now all of this looks extremely strange (especially this extending to be holomorphic across these bizarrely defined cusps). But now I’ll show you why it isn’t strange at all. The matrices that become upper triangular mod ${N}$ (the ones we defined the transformation law for) are a finite index subgroup of ${SL_2(\mathbb{Z})}$ notated ${\Gamma_0(N)}$. The group acts naturally on the upper half plane via linear fractional transformations, and we can form the quotient ${Y_0(N):=\Gamma_0(N) \setminus \mathbf{H}}$.

This is a nice space, but we’d really like it to be a nice compact Riemann surface, so we compactify. Equivalently we have to add in finitely many points. There we have it! These are the cusps. So if we take the extended upper half plane I defined before and then take the quotient we get a smooth Riemann surface ${X_0(N):=\Gamma_0(N) \setminus \mathbf{H}^*}$. Suppose I have some global ${1}$-form on ${X_0(N)}$, i.e. a global section of ${\Omega^1}$. First note that this will be holomorphic across the cusps automatically. The condition that seemed superficial before happens for free when we think in these terms.

Second, this weird transformation law is saying exactly that it is something that can descend to be a ${1}$-form on the surface. Thus a modular form is actually a ${1}$-form on the modular curve ${X_0(N)}$. Or more precisely if we take the quotient map ${\pi: \mathbf{H}^*\rightarrow X_0(N)}$, then a ${1}$-form ${\omega}$ can be pulled back ${\pi^*\omega=f(z)dz}$, and ${f}$ is the “form” satisfying these properties. All of our conditions come out naturally. The cusp forms we were considering were just the ones that vanished at the cusps (the points we threw in to compactify). In algebraic geometry symbols we have ${H^0(X_0(N), \Omega^1)\simeq S_2(\Gamma_0(N))}$. We have to do something a little more subtle to get all our weights, but we’ll come back to that.

Probably the hardest part of breaking into the modular forms literature is that there are so many equivalent ways to think about all these things. We actually need one more interpretation of ${X_0(N)}$ in order to motivate the types of generalizations that happen in the definition of topological modular forms. It turns out that ${X_0(N)}$ represents a moduli functor, i.e. it is a “moduli space”. If you’ve read Mumford’s early stuff on ${\mathcal{M}_{1,1}}$, then you may be able to guess what types of things it parametrizes. It turns out that ${X_0(N)}$ is the moduli space of elliptic curves with certain “level ${N}$” structure. It parametrizes pairs ${(E, C)}$ where ${E}$ is a an elliptic curve and ${C\subset E}$ is a cyclic subgroup of order ${N}$. Two pairs are isomorphic if the isomorphism preserves the subgroup. Surprisingly this moduli space is as nice as can be ${X_0(N)}$.

We could spend months talking about all these modular curves and their properties, but for now we’ll have to leave it at this. Next time we’ll start working towards the generalization.

## 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.

## Taniyama-Shimura 1

It’s time to return to plan A. I started this year by saying I’d post on some fundamental ideas in arithmetic geometry. The local system thing is hard to get motivated about, since the way I was going to use it in my research seems irrelevant at the moment. My other option was to blog some stuff about class field theory, since there is a reading group on the topic that I belong to this quarter.

The first goal of this new series is to understand the statement of the famous Taniyama-Shimura conjecture that led to the proof of Fermat’s Last Theorem. A lot of people can probably mumble something about the conjecture if they have any experience in algebraic/arithmetic geoemtry or any of the number theory type fields, but most people probably can’t say anything precise about what the conjecture says (I’ll continue to call it a “conjecture” even though it has been proved).

The statement of the conjecture is that every elliptic curve over ${\mathbb{Q}}$ is modular. Simple enough, but to unravel what it means to be modular we are going to have to take many posts just for the definition. If you’ve seen this explained before, it might still be interesting to read this series because I’m going to set up the machinery in a slightly different (but equivalent) way so that it will generalize to varieties other than elliptic curves in the future.

We’ll first define modular forms. A modular form of weight ${k}$ and level ${N}$ is an element of the vector space ${M_k(\Gamma_0(N))}$ which consists of holomorphic functions on the upper half plane ${f:\mathcal{H}\rightarrow \mathbf{C}}$ satisfying the additional transformation property that

$\displaystyle \displaystyle f \left(\frac{az+b}{cz+d}\right)=(cz+d)^kf(z)$

for all matrices ${\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)\in SL_2(\mathbf{Z})}$ such that ${c\equiv 0 \mod N}$ (plus something else that we’ll get to shortly).

This is an analytic object if there ever was one. If this is the first time you’ve seen this, then the thing to pay attention to is that these depend on a choice of weight, ${k}$, and level, ${N}$. To get a feel for the level, note that it becomes “easier” to satisfy this transformation law as the level increases, because the amount of matrices we have to check is less. For example, when ${N=1}$ this says our ${f}$ has to behave nicely under every single linear fractional transformation that sends the upper half plane to the upper half plane. One might reasonably guess that ${0}$ is the only holomorphic function with this property. More on this later. The weight is a little harder to get a feel for.

The map ${z\mapsto e^{2\pi i z}}$ is a holomorphic map from the upper half plane onto the punctured unit disk. Note that ${e^{2\pi i z}\rightarrow 0}$ as ${z}$ tends to infinity along the imaginary axis. We can compose with this map and consider our modular form to be a holomorphic function on the punctured disk. This is well-defined because if ${e^{2\pi i z}=e^{2\pi i w}}$, then ${z}$ and ${w}$ differ by an integer and ${f(z+n)=f\left(\left(\begin{matrix}1 & n \\ 0 & 1 \end{matrix}\right)\cdot z\right)=(1)^kf(z)=f(z)}$.

We say ${f}$ extends to be holomorphic at infinity if there is a holomorphic extension to the whole disk. We require modular forms to have this property. Thus a modular form has a Fourier expansion called a ${q}$-expansion denoted

$\displaystyle \displaystyle f=\sum_{n=0}^\infty a_nq^n \ \text{where} \ q=e^{2\pi i z}$

(note that a Fourier series in general involves negative powers, but these would give a pole at infinity). The cusp forms are the subspace denoted $S_k(\Gamma_0(N))$ of the modular forms that vanish at all cusps. To define cusp, just think of the extended upper half plane as ${\mathbf{H}\cup \mathbb{P}^1_{\mathbb{Q}}}$. We stick all the rational numbers along the real line in and also throw in a point at infinity. In practice, we only have to check holomorphic extension across finitely many of these cusps because due to the transformation law we only need to pick on cusp in each equivalence class under the action of the matrix group. When for instance $N=1$ again, all we have to check is that $f$ vanishes at infinity, or upon composing to the disk we get that $a_0=0$.

Do any of these things exist? Well, as we’ve already noted, for small N it seems very hard to satisfy these properties. In fact, our guess was right, $\dim_\mathbb{C} S_k(\Gamma_0(1))=0$ for $1. So until we bump the weight up to 12, we actually only have the 0 function satisfying our properties. For weight 12, there is only one up to scalar multiple. This doesn't look good, but actually when we allow the level to grow we get a lot (even of low weight). But before next time, just ponder how severe the symmetry condition we are imposing is. Somehow every elliptic curve is closely related to one of these which is why the result is so surprising.

Now we have our basic analytic object of the conjecture. The next several posts will go back to the algebraic side of things. Depending on how much detail I decide to give to define the terms in the Taniyama-Shimura conjecture could take anywhere from 4 to 8 or so posts, just to give you an idea of how long you have to hold out for the statement.