This fairly straightforward result (the first part is the fact that orientation preserving isometries of the hyperbolic plane are biholomorphic and so define a complex structure, and the second is from the uniformization theorem) is why the moduli of curves (complex structures on a surface) is just the moduli of marked hyperbolic surfaces (Teichmüller space) after getting rid of the marking (quotient by the mapping class group).

So now, we’ll start looking at the geometry of Teichmüller space. One of the main tools there is the notion of a *quasiconformal map.*

**Definition**: Let and be Riemann surfaces. Let . Then a map is -quasiconformal if is a homeomorphism with continuous differential of maximal rank outside of a finite set of points, and we assume that outside of , with and .

What this means is that away from , the differential takes a circle to an ellipse in the tangent spaces, and the ratio of the lengths of the axes is bounded by .

So what can we do with this? Next time, we’ll really get into a big application of quasi-conformal maps and quadratic differentials, but now, we’ll just look at a couple of basic properties.

We can use quadratic differentials to build some special quasi-conformal maps. Let be a quadratic differential. It gives us a singular Euclidean metric, and we can scale it. In the horizontal direction, we scale by and in the vertical direction by , which gives us a quasiconformal map with constant , and we call these maps Teichmüller maps with initial differential and stretch factor , and they’re of constant area.

This lets us compute:

**Theorem**: Let be a quasiconformal map which is homotopic to a Teichmuller mapping with initial differential and stretch factor . Then where is the derivative of in the horizontal direction and the norm is with respect to the singular metric from .

This can be used to classify annuli. An annulus is where for all . Then for , the smallest quasiconformal dilation is .

This, then, implies that if and only if .

We’ll close with a theorem of Wolpert:

Let and be hyperbolic surfaces and let be a -quasiconformal map. Then for any simple closed curve , we have , where is the length as we discussed before. So that gives a nice interpretation of the in quasiconformal maps: it’s a bound on how much simple closed curves change their length.

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So we start with a definition, as we so often do:

**Definition**: A quadratic differential is a section of , the tensor square of the holomorphic cotangent bundle. It can be written locally as where is holomorphic.

These, in some sense, generalize holomorphic differentials (also called abelian, due to Abel’s study of their integrals) because the square of holomorphic differentials are quadratic differentials (in fact, this route leads us to the Veronese map, but that’s a topic for another day).

So, a quick calculation using Riemann-Roch tells us that the space of quadratic differentials is dimensional on a compact Riemann surface. This is not a coincidence! Some other time, we’ll connect the quadratic differentials more directly to deformation theory and show that this space is naturally dual to the tangent space to moduli. For now, though, let’s see what we can get more immediately.

I dodged talking about foliations with respect to pseudo-Anosov maps last time. This time, though, I can talk about them briefly with respect to quadratic differentials. Let be the zero locus of a quadratic differential . Then, away from , our quadratic differential is a quadratic function on each tangent space, we get line fields given by the vectors on which is positive and negative, called respectively horizontal and vertical.

In fact, these lines are horizontal and vertical lines in a distinguished coordinate: let be a nonzero of , then locally near , we have that is nowhere zero, then we can take any coordinate at and take a branch of the square root to write on some neighborhood containing . Then, we can build a coordinate, evaluated at a point near , by

This coordinate is unique, up to translation and multiplication by , such that . And it’s this coordinate that makes the foliation lines parallel to the real and imaginary axes.

We’ll finish off by noting that the singular metric given by is of finite area, so we actually get a norm on the space of quadratic differentials (which let’s just establish the notation for, though it’s often denoted by ) given by .

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So, on a torus, we classified elements of the mapping class group. Elliptic elements will remain the things of finite order, so we’re going to focus on parabolic elements, which become Dehn twists, and hyperbolic elements that become pseudo-Anosov maps.

**Definition**: A *Dehn twist* is a map that fixes the surface outside of an annulus on the annulus lifts to on *.* The isotopy class of this map only depends on the isotopy class of the embedding of the annulus into the surface.

A Dehn twist is particularly simple on homology, it takes a class to where is the class being twisted and is the intersection number. Denote the Dehn twist around a curve by .

Dehn twists are absolutely fundamental, in fact:

**Theorem**: The mapping class group of a surface is finitely presented, and the subgroup that does not permute the punctures is generated by finitely many Dehn twists on nonseparating curves.

Furthermore, the relations in the subgroup generated by the Dehn twists are generated by:

- Dehn twists on disjoint curves commute
- If and intersect exactly once, then , the Braid relation
- If is a separating curve that cuts off a torus, and are curves in the torus that intersect exactly once, then .
- The Lantern Relation

Unfortunately, it would take me another post or two to properly define everything needed to say what a pseudo-Anosov map is, but here’s the definition:

**Definition:** A map is pseudo-Anosov is it is isotopic to a map with a pair of measured foliations that can be realized transversely and with the same singular points, so that preserves each one and multiplies the transverse measure on by and by .

On the other hand, I can appeal to a big theorem that I also won’t prove!

**Nielsen-Thurston Classification Theorem**: For every , one of the following holds:

- has finite order
- There is a system of disjoint essential simple closed curves fixed (up to isotopy) by
- has a pseudo-Anosov representative.

The first two aren’t mutually exclusive, but anything that isn’t one of them must be pseudo-Anosov, and I’ll just leave it at that, and move on. Next time, back to Teichmüller space.

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Let be a topological surface. Then it has an astoundingly large group of homeomorphisms. This is WAY too big, and has infinitely many connected components. So first, we restrict to the homeomorphisms that preserve orienations, , and then we mod out by isotopy, that is, continuous families of homeomorphisms. So we’re looking at the set of connected components of . We will denote this group equivalently by , by and by where is the genus of and is the number of punctures.

For a moment, let’s just notice that this is an extremely robust notion. Though there’s work required, as long as we restrict to punctured, oriented surfaces, we can replace homeomorphisms with diffeomorphisms and isotopies with homotopies and we get the same group. However, there are two changes that happen occasionally that do change the group, albeit in small ways:

- If we forget about the orientation and allow it to be reversed, then we get an index two extension compared to the above group
- If we allow boundary discs instead of punctures, things get a little bit more complicated. If we restrict to homeomorphisms and isotopies that fix the boundary pointwise, then we end up with an abelian extension of the above group by Dehn twists (mentioned last time, but more on them below) around the boundaries.

For the very simplest surfaces, we can actually compute the mapping class group very explicitly. For the sphere with at most three punctures, as well as the torus with no punctures, we can get really explicit (though the proofs are long and technical, and thus serve little point in a blog post!)

Note that is the sphere with one puncture. Both it and the sphere have trivial mapping class groups. This is essentially because we can take a circle, apply a homeomorphism, and then isotope so that the circle is back to where it started. A hint in the direction of the proof is to use the Jordan Curve Theorem and Alexander’s Trick.

The twice punctured sphere is the annulus, and it turns out to have a single nontrivial element in its mapping class group, consisting of imagining the annulus as a right cylinder and rotating it 180 degrees around a diameter of the circle. Essentially, what this does is it swaps the two punctures. A similar story happens with the pair of pants, which is the three times punctured sphere. It turns out that , the permutations of the punctures. For , things get more complex, and this essentially reflects the fact that (as we will see) the moduli space for is positive dimensional.

Now, before we talk a bit about the specific types of elements of mapping class groups in general (next time! This post is already getting a bit longer than intended), we’ll talk about the mapping class group of the torus.

In general, we get a natural map because isotopically trivial homeomorphisms act trivially on homology. This preserves the intersection form, so when there are no punctures, we get . If , then we have an isomorphism . (And a side note, personally, this map is essentially the exact same thing as the Torelli map taking a curve to its Jacobian, and so is closely related to the Schottky problem, on which I wrote my thesis)

So, we want to analyze this map to compute , and it will turn out to be an isomorphism. Surjectivity follows because any element of acts on , but preserves , and so acts on the torus, and because it’s invertible, this is a homeomorphism. Injectivity isn’t really much harder, it can be done by lifting the question to and showing that any map that is the identity on homology must be isotopic to the identity.

So now that we can describe the mapping class group of a torus, there are three types of elements to look at:

- Elliptic: An elliptic element is any element of finite order. These are very rare, only corresponding to rotation of the square lattice, the hexagonal lattice, the powers of these maps, and their conjugates. These have trace of -1, 0 or 1.
- Parabolic: These elements are also called twist maps, and generalize to Dehn twists that keep coming up. These maps preserve (up to isotopy) exactly one simple closed curve on the torus, and they have trace 2 or -2.
- Hyperbolic: The hyperbolic elements, also called Anosov, have trace outside of , and this is where things get interesting, which means we won’t go too deep. They have two foliations by straight lines (well, on the plane covering it), and they’re
*transverse invariant foliations*, one of which has lengths along it expanded by the element, and one of which is contracted.

Secretly knowing the answer to what the dimension of the moduli space is (over the reals, it’s ), I know how many coordinates we need to construct. Half of them come immediately from the discussion last time of pairs of pants:

**Theorem**: For any triple of positive numbers , there exists a unique, up to isometry, hyperbolic pair of pants with boundary circles of length , and .

This is actually not a hard theorem to prove. It suffices to show that there is a unique, right angled, convex hyperbolic hexagon with three, non-pairwise-consecutive sides of lengths . Then take another copy and glue them together along the other sides. The existence and uniqueness of such hexagons is a pretty straightforward construction in hyperbolic geometry, just like similar things in Euclidean geometry are.

Now, we fix a pair of pants decomposition of a hyperbolic surface. Last time, we said that you need curves to define it, and their lengths provide us with coordinates , but that’s not quite enough to reconstruct the hyperbolic surface completely.

To try to build the surface, we’ve currently just got a pile of pairs of pants, we need to figure out how to glue them together. We could just identify them by setting the seams (remember that we got them by gluing together hexagons) to be a single curve, but that’s not the only way. We can still rotate them with respect to each other, and we get isometric surfaces every time the rotation parameter changes by .

However, we don’t want to try to directly parameterize hyperbolic structures, we actually need more data. A *marking* of a hyperbolic surface is a homeomorphism preserving orientation from a topological surface , and a pair of marked hyperbolic surfaces are only equivalent if there’s an isometry such that and .

So it turns out that for *marked hyperbolic surfaces* there isn’t this periodicity. When we go through a whole turn, the marking has changed by what’s called a *Dehn twist*, and we’ll talk about them in more detail later. But the point is that for marked hyperbolic surfaces, we can actually distinguish the gluings by more than just the angle, and this gives us some more coordinates.

**Theorem**: For each pair of pants decomposition of , each point of is associated to a marked hyperbolic surface with described lengths and twist parameters. Furthermore, every marked hyperbolic surface occurs this way.

These are called the *Fenchel-Nielsen Coordinates* and the space of marked hyperbolic surfaces is called *Teichmüller Space*. Topologically, this space, which we’ll denote , is very simple: it’s a product of copies of the real numbers with the positive real numbers, and so is contractible!

We’ll talk just a bit about the topology before calling it a day, though. We can actually define the topology on directly, and it turns out to be the same. Let be a marked hyperbolic surface and (just typing this feels like I’m breaking a self-imposed rule, but I swear, we’re not about to go into the jungle of estimates and asymptotics, though they certainly relate here! Mostly because I have other things to talk about) we can define as the set of surfaces such that for every simple closed curve $c\subset S$, we have . These manage to be a basis for the topology induced by the Fenchel-Nielsen coordinates.

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**Definition**: A *hyperbolic surface of genus g* is a topological surface of genus g along with a metric that is locally isometric to the hyperbolic plane. Equivalently, it has a Riemannian metric of constant curvature -1.

There are some distinguished types of curves on a hyperbolic surface, and I don’t just mean the geodesics (though we’ll relate them to geodesics). This is going to be a definition heavy post, but hey, we’re doing a construction, this tends to happen.

**Definition**: A closed curve is peripheral if it freely homotopic to a boundary component of the surface. It is essential if it is neither peripheral nor contractible, and we will ALWAYS assume that closed curves are essential, because otherwise they don’t give good homotopy classes. A closed curve is simple if it is the image of an injective map .

A key observation, that isn’t particularly difficult, is that each free homotopy class of curves on has a unique closed geodesic in it. So we will speak of THE geodesic representing a free homotopy class.

It’s also not so hard to see that if is a simple closed curve, the geodesic representing its class is also simple, and if represent different free homotopy classes, both are simple, and they don’t intersect, then their geodesic representatives don’t intersect. So we can reduce a LOT of questions to just looking at geodesics.

So a procedure we can do is take any simple closed curve, replace it by the corresponding geodesic and then cut the surface along that geodesic, getting a new surface with two geodesic boundary components. This can happen in two ways, either it can separate the surface into two surfaces, whose genera add up to the genus of the original and each with one extra boundary component, or else it can decrease the genus by one, but increase the number of boundary circles by two. Remember these operations, we’ll see them again in other forms many, many times in the study of moduli!

So what are good ways to cut? Well, the simplest possible hyperbolic surface is a sphere with three boundary components, which we call a pair of pants

Why is it the simplest one? There are no essential curves! Every simple closed curve on the pair of pants is either freely homotopic to the boundary or is contractible.

So what we do is we take a maximal set of disjoint simple closed geodesics on our surface, and that will give us a pair of pants decomposition. A compact surface has Euler characteristic and a pair of pants has , so that tells us that we need pairs of pants to decompose a genus closed surface, and that means we need geodesics. (for those with any familiarity with this stuff, these numbers should look eerily familiar…and that’s absolutely no coincidence.

Let’s finish by looking at the *combinatorial type* of the pair of pants decomposition, which will be very similar to some things from dual graphs of stable curves that we’ll get to in 2015 at some point. The combinatorial type is a graph with vertices for each pair of pants and an edge for each boundary component connecting the two components that it’s the boundary of. Any trivalent graph with vertices arises as the combinatorial type of some pair of pants decomposition of a genus closed surface.

Next time, we’ll talk about coordinates and actually build Teichmüller space itself.

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We actually have to start with some more basic differential geometric notions that have been neglected, because we never did a “Differential Geometry from the Beginning” series (perhaps I should in the future? Chime in in the comments if that would be desirable, though I might try to find someone else to write it, as I’m not a real expert on that stuff). We’ll just assume, though “smooth manifold” and “Riemannian metric” largely because they would deserve posts of their own, and these are things that I think the target audience of these posts are probably ok with (correct me if I’m wrong!).

**Definition** The hyperbolic plane (in the Poincaré half plane model) consists of the points of with and which has metric . This means that if you start from any point, the real line is infinitely far away from it.

A quick historical note: even the Wikipedia page points out that the half-plane model is due to Beltrami, a name that reappears often in Teichmüller theory.

So, I should include a picture, and these are some geodesics in the hyperbolic plane:

So what sorts of isometries are there? The linear fractional transformations! If we treat as , then we can write them as where is a matrix with determinant 1. This group is precisely . In fact, the correct group is because minus the identity acts trivially.

As a final aside on the half-plane in and of itself, we look at the subgroup , and we can see that it acts very nicely on the upper half-plane (though not quite as nicely as you’d really hope for quotients, and that’s part of the fun) and some finite index subgroups act even more nicely. We call this the modular group (and some of the subgroups are called congruence subgroups) and the quotient of the upper half-plane by it is the modular curve (or a modular curve). Modular curves are VERY important, and show up in all sorts of problems in number theory (which I’m sure Jim will talk about at some point), including the proof of Fermat’s Last Theorem.

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**Books**

*Algebraic Curves and Riemann Surfaces* by Miranda*
Geometry of Algebraic Curves Volumes I and II* by Arbarello, Cornalba, Griffiths and Harris

**Papers
**

“Complete subvarieties of the moduli space of smooth curves” by Diaz

“Curves and their moduli” by Harris

“Algorithms for computing intersection numbers on the moduli space of curves, with an application to the class of hte locus of Jacobians” by Faber

“The structure of the moduli spaces of curves and abelian varieties” by Mumford

“On the Kodaira dimension of the moduli space of curves” by Mumford

“Picard groups of moduli problems” by Mumford

“The irreducibility fo the space of curves of a given genus” by Deligne and Mumford

“The projectivity of the moduli space of stable curves I: Preliminaries on “det” and “Div” ” by Knudsen and Mumford

“The projectivity of the moduli space of stable curves. II. The stacks ” by Knudsen

“The projectivity of the moduli space of stable curves. III. The line bundles on , and a proof of the projectivity of in characteristic 0.” by Knudsen

“The Picard groups of the moduli spaces of curves” by Arbarello and Cornalba

“The second homology group of the mapping class group of an orientable surface” by Harer

“The virtual cohomological dimension of the mapping class group of an orientable surface” by Harer

“Divisors in the moduli spaces of curves” by Arbarello and Cornalba

“Teichmüller space via Kuranishi families” by Arbarello and Cornalba

From *Moduli Spaces of Riemann Surfaces* edited by Farb, Hain and Looijenga:

“Teichmüller Theory” by Hamenstädt

“The Mumford Conjecture, Madsen-Weiss and Homological Stability for Mapping Class Groups of Surfaces” by Wahl

“Lectures on the Madsen-Weiss Theorem” by Galatius

“Tautological Algebras of Moduli Spaces of Curves” by Faber

I’ll update this list when new references are introduced, should I do so.

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Of course the paper is about the torsion in certain kinds of elliptic curves. The landmark result in this direction is of course Mazur’s theorem, stating that if is any elliptic curve over then the torsion subgroup of is isomorphic to an element of the following:

Moreover each group occurs as a torsion subgroup. There is an exercise to this effect in Silverman’s Arithmetic of Elliptic Curves. Merel’s uniform boundedness theorem tells you that for any number field there is a similar but likely larger “Mazur-style list” depending only on . This is to say for each positive integer there is a list of groups such that there is a number field of degree and an elliptic curve with that torsion subgroup, and conversely the torsion subgroup of any elliptic curve over lies in that list.

The bad news is that when we consider all elliptic curves, it’s pretty hard to come up with these lists. A complete list is only known in degree 2 thanks to work of Sheldon Kamienny, Monsur Kenku and the late Fumiyuki Momose in the late 80s and early 90s. For higher degrees (up to 7), this is a nice accounting of the details.

The good news is that we can get a much better idea of what the torsion really is with a subset of elliptic curves. For any elliptic curve the set of maps of complex Lie groups contains a copy of the integers. If it contains anything else we say that has *complex multiplication* or CM.

The answers here are simpler because the Galois representation of is much smaller. But at the same time, CM elliptic curves capture lots of extremal behavior. For instance, if is a prime, work of myself, Clark, Cook and Rice shows an instance of this for -torsion points of elliptic curves over number fields.

Myself, Clark, Corn and Rice were able to compute all possible torsion subgroups over number fields of degree (it was known up to degree 3 previously, please see our paper for details). Despite the fact that this was just published, this paper has been in the works for several years. We and others noticed that torsion over prime degree number fields got quite sparse.

For context, the part of Mazur’s theorem for CM elliptic curves was previously known by a theorem of Loren Olson, stating that the only possible torsion subgroups up to isomorphism over are what we grew to call the “Olson groups:”

Schuett was one of those who noticed that for , the Olson groups were the only possible torsion of a CM elliptic curve over F. He asked if it might be that this is pattern continues for all large primes. Well, that’s exactly what myself, Bourdon and Clark just proved this past year. Moreover, over *all* prime degree fields there are only 17 isomorphism classes of CM elliptic curves with a torsion subgroup besides one of the Olson groups.

Now of course a new idea is needed for something like this. The key is a field called the CM field of E. The idea that we came upon is that usually if and is a CM elliptic curve with then . Moreover if is coprime to the discriminant of the ring of maps over the complex numbers, this embedding is not an isomorphism.

This is something that should be surprising! Cyclotomic fields and fields generated by CM elliptic curves are two very distinct types of extensions of the rational numbers! For instance, genus theory says that the intersection of a cyclotomic field with the field generated by the -invariant of a CM elliptic curve is multiquadratic. So there’s a fundamental tension here.

We were not able to show that the CM field argument works for all fields and all , but for number fields with a real embedding, we proved it. In particular, this works out whenever is odd, and in particular for large primes.

The point of having a real embedding means that we can use the action of complex conjugation – and we know how that acts on a CM elliptic curve over a real number field because Gauss’ genus theory ALSO tells us about the models of a CM elliptic curve over the real numbers.

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