**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.

Filed under: Uncategorized Tagged: hyperbolic geometry, hyperbolic surfaces, teichmüller space ]]>

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.

Filed under: Uncategorized Tagged: hyperbolic geometry ]]>

**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.

Filed under: Uncategorized Tagged: moduli space, riemann surfaces, teichmüller space ]]>

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.

Filed under: Uncategorized Tagged: Bourdon, Clark, complex multiplication, elliptic curves, preprint ]]>

Filed under: Uncategorized ]]>

I’ll keep my remarks brief here, but roughly a Mori Dream Space is an algebraic variety which “looks like” a toric variety, and so its birational geometry is strongly determined by some combinatorial invariants. There’s obviously much more to be said here, but for that I will defer to either the papers listed above or to a survey on Cox rings such as this one. The Mori Dream Space conjecture for a family of varieties is simply the statement that for all , is a Mori Dream Space. What Castravet and Tevelev showed was that for , there are Birational maps between and spaces which are not Mori Dream Spaces. I haven’t yet had a chance to do more than skim Fontanari’s paper.

One thing I can note here is that the object you most often study to get a handle on the birational geometry of is the cone of nef divisors. I myself had a paper with Arap, Gibney and Swinarski studying a certain selection of these divisors called Conformal Blocks Divisors, which can be defined either on or . One of the things that made this conjecture difficult experimentally was that the number of generators of this nef cone gets very big very quickly, and the work with Conformal Blocks was in part an attempt to focus on some subcone where we could actually do experiments. It might be interesting to do some more experiements and see what exactly goes wrong with the simple quotient under .

Finally, even though a fly may drop, it may also start buzzing again, as noted by Ellenberg. Still, interesting stuff!

Filed under: Algebraic Geometry, Uncategorized ]]>

As for “Yay” (cue youtube), the biggest reason for the “Oops” is that my thesis is finally posted to the arXiv! The next project won’t take so long.

Filed under: Uncategorized ]]>

However, for today, we’re going to take something most of us first saw in differential geometry (I first met this map in do Carmo‘s book) and translate it into algebraic geometry.

We will start in the absolute least general way possible, following do Carmo. Let be a surface. Then there’s a map to the unit sphere taking each point to its unit normal vector. This is the Gauss map, and it’s a REALLY useful tool, as anyone who has gone through this book can attest. For instance, if you want to define the curvature of a surface in , the Gauss map is essential. For instance, the Gaussian curvature is the determinant of the derivative of , and in fact it would be redundant to go through everything about the Gauss map for surfaces in because there’s a whole chapter in do Carmo titled “The Geometry of the Gauss Map!”

We’re going to generalize and then algebraize. First, let’s just drop the orientation on our surface. To forget that, we can replace the normal vector with the normal line. So then instead of getting a point in we get a pair of antipodal points, or just a point of from our surface. Then we can see that the map is really given by taking the inclusion, then we have , and then taking the line perpendicular to the image plane. Taking the union of these maps, we just have the map . Then, using the Riemannian metric on , we can make this a map , and follow it up with the fact that , and project down, to get the map , and then we can rewrite it by taking each point to the line in , which gives us a map , giving the usual Gauss map.

So how can we simplify and generalize this? Step 1 is to replace the normal vector with the tangent space, which gives a point in the dual projective space. Then we want to generalize dimension. It’s easy to handle hypersurfaces in , we just get a map to (or rather, to its dual). In general, if we allow non-hypersurfaces, we get maps to Grassmannians, so gives us a Gauss map .

Now, we’re going to let the target space vary. We just need a space such that where is the tangent space at some specific point. If we have trivial tangent bundle, we can identify all the fibers and then the derivative of our map actually gives us a map from the domain to a Grassmannian. What are some spaces that have this property? Lie groups! It’s important that we have Lie groups, not just homogeneous spaces, because of the *unique* way that we can identify fibers.

Now, if we try to algebraize, the first thing we get is a Gauss map for affine varieties . We can even get rid of and replace it with an algebraic group , but most of the algebraic groups that immediately come to mind are affine, things like etc, the classical groups. Plus and and products of these. But that still only gives us affine varieties, nothing projective or complete. Fortunately, there’s one remaining option that are commonly studied: abelian varieties. Though we can’t do much with rational varieties (as there are no maps from them into abelian varieties other than constant maps), we can get a lot of mileage out of the Gauss map on abelian varieties, as we’ll see in the next post.

Filed under: AG From the Beginning, Algebraic Geometry, Complex Analysis, Differential Geometry ]]>

We can use to construct a lattice. Inside of , there’s naturally a lattice , but it’s not of full rank, the quotient isn’t compact. However, if we add in another rank lattice that is independent from the natural one, that will be full rank. So now, we take the lattice . This is then a lattice in of full rank that varies holomorphically with the pair of Riemann surface and symplectic basis of .

This means that the quotient varies holomorphically with that data. In fact, because a change of basis is linear, the transformation extends to all of , and so the quotient doesn’t depend on which symplectic basis we chose! Thus, this torus, which we will denote by only depends on the Riemann surface, not on the symplectic basis.

We can also define a function on out of , and it will behave decently with respect to the lattice. Define , the Riemann theta function, to be the multivariate Fourier series . It’s not hard to see from this definition that if we translate by an element of , the function is invariant. And it’s holomorphic (convergence is guaranteed because is symmetric and has positive definite imaginary part) everywhere. So it can’t be periodic in the other directions. In those, you pick up an exponential factor.

Because of this, is periodic, and so we get a divisor on called the Theta divisor, . The geometry of this divisor is a rich and detailed subject of study, and we’ll talk about it a bit in later posts. For now, the main point is that for any Riemann surface , this divisor defines what is called a principal polarization. One way to see that is because the bilinear form on the lattice is unimodular. Another way is by directly computing that has a unique (up to scaling) global section. The third, and in many ways most geometric, way is by looking at the map given by where is the translate of by . This gives an isomorphism between and the dual torus of degree zero divisors on .

Next time, we’ll have an interlude with an application of the geometry of the theta divisor, and then we’ll get back to constructing things with 1-forms on curves.

Filed under: Abelian Varieties, AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves ]]>

Today’s word is one that I use quite a bit: algebraic variety. The kanji is 代数多様体. The pronounciation is “daisutayoutai.” Really this works out as “algebraic manifold.”

In fact, 代数 appears to be a prefix that means algebraic, and is used in many other situations. Here, the two kanji are actually pretty straightforward in why they make something algebraic.

The first is 代, which here is pronounced dai, and has a great many other pronounciations. It also has a great many different shades of meaning. But the obviously relevant one here is “substitute” or “replace.”

The second kanji, 数, is pronounced ~~kazu~~ su here, again among many other things. The meanings are “number,” “law,” “figures,” and the like. (According to a commenter, only “number” and “to count” are common meanings, the others may be archaic)

So together, they mean something like “substituting numbers” or “laws of substitution,” which mean algebra. As last time, we discussed the word “tayoutai,” or manifold, this means that daisutayoutai translates as “algebraic manifold.”

Filed under: Japanese for Mathematics ]]>