 This talk will be about the Riemann-Roch theorem in the case of genus two surfaces. So when G is equal to two, the Riemann-Roch theorem states that the dimension of the space of functions with poles only on D is the degree of D plus one minus G plus L of K minus D. And of course, since G is two, we can replace this term by minus one. And we also know as usual that the degree of K is two G minus two, which is two, and L of K is G, which is equal to two. Now what we want to do is to try and work out what L of D is in terms of degree of D. So first of all, suppose the degree of D is less than zero, then we know that L of D is equal to zero as always. If the degree of D is equal to zero, then as we've seen before, the dimension of L of D is either not or one, and it's one in the case when D is equivalent to zero and zero otherwise. And if the degree of D is greater than two, then L of D is equal to the degree of D minus one, because the degree of K minus D is now negative, so L of K minus D is zero. If the degree of D is equal to two, then L of D is equal to zero or is equal to one or two, and it's equal to two if D is equivalent to a canonical device as you can check easily and one otherwise. This is essentially dual to the result when the degree of D is zero. If the degree of D is one, then elementary arguments show that L of D is zero, one or two, but it can't be equal to two, because if it were two, we would have a two-dimensional space of functions and this would give us a rational map to the projective line, which was generically one to one, which is not possible because that would mean the surface of genus zero. So the degree has to be zero or one. And this means that we can draw a sort of graph of what L of D looks like as follows. So I suppose we just draw minus two, minus one, zero, one, two, three, so this is the degree of D, and up here we have the dimension of the functions with poles only on D, and the graph sort of looks like this, except we've got these ambiguities because for when degree of D is zero, this could be zero or one, here it could be zero or one, and here it could be one or two. So really the graph sort of looks like one of these things here. And just for comparison, you remember when the genus is zero, the graph looks like this, and when the genus is one, graph looks like this, except that you remember there was a bit of an ambiguity here because it could go like that instead. So the canonical divisor is here for genus two and here for genus one and here for genus zero. And these points are the ones where the divisor is equivalent to zero, and these points are the ones where the divisor is equivalent to a point. So you see as the genus goes up, we get more and more ambiguities about how the dimension of L of D varies. Next, we can look at some examples of genus two surfaces. So we can take the following examples. We can take the curve Y squared equals X minus alpha one, X minus alpha two, X minus alpha three, X minus alpha four, X minus alpha five. Or we can take the curve Y squared equals X minus alpha one, and can't be bothered to write it out, up to X minus alpha six. And both of these, if they're compactified, will give us examples of genus two surfaces. You can see they're all two to one maps from the curve to the projected plane, if we don't worry too much about the point of infinity. So this is just taking a point X, Y to X. And it's two to one, except at these, segment X is one of these six points, possibly including infinity, where there are two possible values of Y. So we can draw a picture of the surface as follows. We take two copies of the projective line, and I'm gonna draw the projective line in a slightly funny way like this, because what I've done is I've taken the points alpha one, alpha two, alpha three, alpha four, alpha five, and the point either alpha six or infinity, depending on which of these you're doing. And I've cut the projective line along here. So I've made a cut from alpha one to alpha two, and from alpha three to alpha four and so on. So now there's a sort of circular bitch. And I'm going to do the same thing with the other copy of the projective line. And we get something like this. And now we should glue these together like that. So we get a sort of surface which looks a bit like this. You see, it's really a genus two surface, which is a sort of sphere with two handles on. So this shows that these curves do indeed have topological genus equal to two. We can also look at one forms on these surfaces. And for hyper elliptic surfaces such as these, we can easily write down one form. So we have y squared equals x minus alpha one, I hope to say x minus alpha six. Hyper elliptic just refers to curves of the form y squared equals polynomial in x. In the case when it's a polynomial of degree three or four, you get an elliptic curve. And we've got some obvious differentials, which are dx over y and x dx over y. And these are actually holomorphic. Well, we need to check the holomorphic at the points where y is equal to zero and also at the point infinity. And at y equals zero, we just use the fact that two y dy is equal to this rather monstrous expression. We take x minus alpha one up to x minus alpha five plus x minus alpha one to x minus alpha four times x minus alpha six plus various other expressions times dx. And the only thing you really need to know about this mess is that it's not equal to zero at x equals alpha one up to alpha six, as you can easily check. And then we see that dx over y is equal to two y dy over y times this mess, which is equal to dy over something that's none zero. And this is regular at x equals alpha i. So it doesn't have a pole at y equals naught. We should also check what happens at x equals infinity. So let's put z equals one over x and when x tends to infinity, the term y over x cubed tends to a finite value. So let's put w equals y over x cubed. And this equation then becomes w squared equals one minus alpha one z up to one minus alpha six z. And we find that x to the n dx over y is equal to minus c to the one minus n dz over w. Well, w is quite harmless because it's finite none zero at z equals naught. But we see from this that we must have n less than or equal to one in order for this thing to be polymorphic. This is why things like x cubed dx squared dx over y are no good because they have a pole at the point at infinity. So we just get two dimensional space of one forms on this hyper elliptic surface. In general, if I went up to x to the eight, then we could have an x cubed here, but not an x, sorry, an x squared here, but not an x cubed and so on. Next, we can look at the possible zeros of a one form. So if we've got this hyper elliptic curve, y squared equals x minus alpha one up to x minus alpha six, you can sort of think of it as something like this. So here's alpha one, alpha two and so on. We've got six points and the curve is going to look something like this, maybe. And the points generally come in pairs with the same x coordinates. So we've got various pairs of points. And if we look at a typical one form, which would be something like a plus bx over y dx, this will have two zeros at x plus or minus y. We'll have zeros at a pair of points or it might have a double zero at some point alpha i zero. So the zeros of a one form always has two zeros and these are usually at distinct points, but there are six special points where it has a double zero. So these are actually, there are six points on this curve where a one form has a double zero. And this means that these six points are not equivalent to any of the other points on the hyper elliptic curve under an automorphism. So this is quite unlike the case of genus zero or genus one curves. For instance, on a genus one curve, the group of automorphisms is transitive so there aren't any special points. Whereas as soon as you get to genus two and above, you find lots and lots of rather special points. So these points can sometimes called theta characteristics. So a theta characteristic is roughly a divisor d such that two d is equivalent to the canonical divisor. In other words, it's a set of zeros of a one form. So you see, if you take each of these points, there's a one form with a double zero at that point. So two times this point is equivalent to the canonical divisor. So these are some examples of theta characteristics. From this, we can also work out the automorphism group of an elliptic curve, so hyper elliptic curve. First of all, it's got a group z modulo two z taking y to minus y, remember y squared is equal to x minus alpha one up to x minus alpha six. Secondly, you may be able to permute the values alpha one up to alpha six. And these are all elements of p one and these can be permuted using the group pgl two of c. Which is the automorphism group of p one. And what's the size of the subgroup of pgl two c I'm using these points where it depends what points you choose. Usually that there won't be any permutations of these in pgl two c. So the automorphism group of the hyper elliptic curve will have order two, but there are some cases when it can be much bigger. For example, if we take the numbers alpha i to be zero one i minus i minus one and infinity, you can think of these as being the vertices of an octahedron in the Riemann sphere. Then it's got various permutations. For instance, you can take x to i times x or x to x plus one over minus x plus one. And these will generate a group of permutations for these of order 24. In fact, it's the group of rotations of an octahedron. So the automorphism group of the corresponding hyper elliptic curve y squared equals x to the five minus x. If you compactify it at a point at infinity, then its automorphism group will be zero over two z times the group of rotations of an octahedron which is the symmetric group S four. In fact, this is the maximal possible order of a hyper elliptic curve. Of course, this affine curve has a smaller automorphism group than that because the pointed infinity would be fixed. The automorphism group is only this big if you compactify this by adding a pointed infinity. Now we can try and identify, we want to show that all genus two curves are in fact hyper elliptic. And to do this, we can study the size of NP for P being a point. So how can this behave? Well, suppose we have N down here. It can be minus two, minus one, zero, one, two, three and so on. What are the possibilities? Well, obviously it has to be zero if N is less than zero. And for N equals zero, it's obviously one because we've got the constants and here it must be one because the curve is not a genus zero. And then we've got an ambiguity here and then carries on like that. So there are two possible ways this function can behave. And you can see that this happens when two P is equivalent to the canonical divisor. And this thing happens otherwise. So there are six points where this happens and all the other points, this happens. These points by the way are called Weierstrass points. So Weierstrass point is a point P where L of P is bigger than usual. So you find that generically the most points have L of NP being something fairly small. And occasionally you get points where it's a bit bigger than usual. So for genus zero or one curves, there are no Weierstrass points because L of NP is entirely determined by N. But in genus two and above, we start getting all these rather special points. Anyway, in general from the, by looking, using the fact that canonical divisor has two dimensions, this gives us a map from the curve to P one by taking two elements of this as coordinates on P one. And we're gonna pick P to be a ramification point of C for this map. So this means that an L of two C is now equal to two. So in other words, we're picking P to be one of the Weierstrass points. And let's work out what L of NP looks like and work out what the corresponding functions are. So let's take N to be zero, one, two, three, four, five, six, seven, eight, nine, 10. And what's L of NP? Well, it goes one, one, two, two, three, four, five, six, seven, eight, nine. And you should notice that here, there's a sort of here and here, there's sort of glitches where the number doesn't go up by one as it usually does. So let's try and write down the basis for the functions for all these spaces. Well, for N equals zero, it's obvious what we get, we just get constant functions. For N equals one, we get nothing new because one is equal to that number there. For N equals two, we get a new function, X with a double pole at P. So we don't know what X is yet, let's just call it X. And for N equals three, we see we get no extra functions. For N equals four, we get one extra function with a pole of order four. Well, that's not really new because it's just X squared. Then for N equals five, we get a new function and we can't get that from any polynomial in X. So it's a new function, let's call it Y, which has an order five pole at P. And then when N is six, we want an order six pole, well, we can get that from X cubed. When N equals seven, we get an order seven pole, well, that's X, Y. Then we get an order eight pole, that's X to the four. Then we get an order nine pole, well, that's obviously X squared Y. Then we get an order 10 pole, well, that could be Y squared or X to the five. And at this point, we stop because if you count up, we know one, two, three, four, five, six, seven, eight, nine, 10 functions in a nine-dimensional vector space. So we get a linear relation between them forced by the Riemann-Roch theorem, which says that something times Y squared plus something times X squared Y plus something times X Y plus something times Y equals something times X to the five plus something X to the four plus something X cubed plus something X squared plus something X plus something. Well, we can of course simplify this a bit. So we can change Y to X squared to Y plus something times X squared times Y, if we're not in characteristic two and get rid of this term here. Similarly, we can get rid of this term by adding a multiple of X to Y and we can get rid of this term. And we can get rid of this constant term here by rescaling X, assuming that our field is algebraically closed and so on. So in an algebraically closed field of characteristic zero, we can reduce to the case of the curve Y squared equals X minus alpha one up to X minus alpha five. So over the complex numbers, every genus two curve is of this form. By the way, a bit of a warning here. This is not a non-singular curve in two-dimensional project space. So you remember for elliptic curves, we had Y squared was a cubic polynomial in X and that was regular at infinity. If we choose projective coordinates, for this case, we get Y squared Z cubed equals X minus alpha one Z up to X minus alpha five Z. And if we look at the point zero one zero, which is X, Y, Z, this is a point on the curve and we can look at it locally by setting Y equals one and we find we've got the curve Z cubed equals X minus alpha one Z up to X minus alpha five Z. And now you see this is singular at X equals Z equals zero. So this curve here, if we try and embed it as a plane curve, it picks up a singularity at infinity. For elliptic curves, this didn't matter because we had a Z to the one here, so it was regular at infinity. So how can we embed a hyper elliptic curve as a non-singular plane curve? The answer is you can't because a non-singular plane curve in P two of degree D as genus G minus one, G minus two over two, sorry, D minus one times D minus two over two, which is one of the number zero, one, three, six, 10 and so on. So a curve of genus two cannot be embedded as a non-singular plane curve in the projective plane. Well, if we can't embed our hyper elliptic curve as a non-singular plane curve, how can we represent it? Well, let's just list a few. So these are ways to represent the genus two curve over the complex numbers. So first of all, we can just represent it as a double cover of the projective line branched at six points, which should be Y squared equals X minus alpha one to X minus alpha six, which is the one we've been studying. Now for elliptic curves, we saw we could represent them as the complex numbers modular lattice. So this is G equals one and for G equals naught curves, we could just represent it as the Riemann sphere. And there's an analog for genus two curves, we can represent it as the opper half plane modular discrete group. So what is going on is for any Riemann surface, we can represent it as a simply connected covering space modular discrete group. And the possible simply connected Riemann surfaces or the Riemann sphere, which gives you genus naught surfaces, the complex plane, which gives you genus one surfaces and the opper half plane, which gives you absolutely everything else. And so in this case, we would work with elliptic functions, which are functions on the complex plane invariant under the lattice L. The analog in this case are called automatic functions and modular functions sometimes. This group here is called a Fuxian group. Fuxian group is just a discrete subgroup of PSL2R, which is the group of automorphisms of the opper half plane. And everything you can do with elliptic functions has an analog and it's even more interesting and more complicated for automorphic functions and Fuxian groups. And I'm not going to discuss that because it's more appropriate for an entire lecture course rather than for a few minutes at the end of a lecture. So that gives a second way of representing genus two curves. A third way, you can represent them as plane curves in P2 with one double point. So a double point will look something like that. If you want to represent them as curves without singularities, you can represent them like that in P3, where you can take the intersection of a suitable quadric intersection with a cubic. And if they're intersection of these contains a line, then the rest of it will usually be a genus two curve. So although you can't embed them as non-singular curves in the plane, you can embed them like that in three-dimensional projective space. Finally, we can do an analog of what we did with the elliptic curves. We take a map Z and we integrate the differentials. So for elliptic curves, we integrated from a fixed point A to Z of DX over Y. Well, for a genus two curve, we've got two independent differentials. So we should also integrate X DX over Y. And this will sort of be an element of C squared, except it's not really defined because there are many different paths from A to Z. So there's an ambiguity. Now for elliptic curves, the ambiguity was a lattice generated by two elements. Here the ambiguity is a lattice generated by four elements because the first homology of a genus two surface is Z to the four. So this is isomorphic to Z to the four. This space here turns out to be a projective variety called the Jacobian of our elliptic curve C. So what we have here is a map from our curve C to the Jacobian, which is C squared modulo a lattice and is therefore topologically a torus S one to the four. So we did an analog of this for elliptic curves and it happened to be an isomorphism. So the elliptic curve is actually isomorphic to its Jacobian. For genus greater than one, the Jacobian is bigger than the curve we start with. And instead of getting an isomorphism, we just get an embedding of the curve into its Jacobian. So that gives five ways of representing the genus two curve. Okay, I think that's enough about genus two curves the moment the next lecture will presumably be about genus three curves.