 This algebraic geometry lecture will be about elliptic functions and we will be using them to show that several cubic curves in the plane are not birational to the affine line. The previous lecture proved this using an algebraic method, so this lecture is going to be different because we're going to use an analytic method. So we have to recall the definition of the Weierstrass p function, p of z. This funny squiggle is actually a letter p. It's some sort of weird calligraphic letter p from some 19th century calligraphic alphabet and as far as I know every letter of this alphabet has been lost except for the capital p which was used by Weierstrass for his function and has been copied by mathematicians ever since. Anyway the Weierstrass p function is an elliptic function. So an elliptic function is one such that f of z plus lambda is equal to f of z for all lambda in some lattice l in the complex plane. So a lattice is just something looks something like this. So if this is nought and this is a and this is b then the lattice consists of all integral linear combinations of a and b. So for example a plus b to a and so on. So this function has two periods a and b. Now the function cannot be holomorphic unless it's constant because it's if it were holomorphic it would be bounded in a fundamental domain and therefore bounded everywhere so it has to be constant by Lieuville's theorem. So f has to be meromorphic. It must have poles somewhere. So how can we find functions f with this property? Well one obvious way of finding it is to take any function g and just sum over all lambda in the lattice. Seem to be confused about whether I'm calling it l or lambda. So we sum over all elements of the lattice lambda and if we define f of z like this then it's completely obvious that f of z plus lambda will equal f of z. So what do we choose for g? Well it has to be something with a pole otherwise sorry it has to be something that converges. This means it must decrease reasonably faster at infinity. Well let's try g of z equals 1 over z. Well in that case it's easy to check that this doesn't converge. See the number of lambdas with absolute value less than some constant c grows quadratically in c and this only decreases linearly in c so that's not good enough. If we take gz equals 1 over z cubed this converges well. If we take gz equals 1 over z squared this is borderline by borderline I mean it doesn't converge but it only just fails to converge by the sort of minimum possible amount it could fail but like. What this means is that if we modify this very slightly we can actually get it to converge. So the vice-trice p function is defined like this. We take p of z to be 1 over z squared plus sum over lambda not equal to zero of 1 over z minus lambda squared minus 1 over lambda squared. And what you see we've done is we've just added a sort of constant and the constant is doubly periodic so in some doubly periodic so adding all these constants shouldn't really affect the fact that this is doubly periodic. And the nice thing about this constant is that it's the constant term of this expression here. So z is large then 1 over z minus lambda squared is approximately 1 over lambda squared plus even smaller terms. So what we're essentially doing is adding this sort of this constant sum of lambda does not equal zero of 1 over lambda squared to everything. The trouble is this does not converge so this is not convergent but we're sort of adding this non-convergent expression to this non-convergent expression and producing a convergent expression. So we've got a perfectly well-defined function. There's one slight problem that since we've twiddled everything by adding in this constant it's no longer clear that this is doubly periodic. However this is not very difficult to show because if we take the derivative of this then we're essentially summing all the translates of 1 over 1 plus c cube which is convergent up to some constant derivative of this is 1 over z cube not to a constant that should be a 3. So the derivative of this is doubly periodic and this means that this function here is doubly periodic up to a constant. So we find that rho of z plus a is equal to so not rho p of z plus some constant and this constant must be zero because this function here is an even function. So the fact this is even implies this is constant. So this slight change the definition to make this convergent fortunately doesn't affect the fact that this is still doubly periodic. You've got to be a bit careful with this argument because there are some quite similar functions where you do this trick and adding in this constant actually stops it from being periodic but anyway we've got a nice doubly periodic function. Next we have a look at the Laurent expansion of p by strice p function at z equals zero and this isn't very difficult to work out it's 1 over z squared and the constant term vanishes so we get naught times z plus something times z squared and we don't really care all that much for what this is. So we can look at the derivative well this is going to be minus 2z to the minus 3 plus again the constant term is going to vanish and we're just going to get a linear term something in z. Now we take the derivative and square it well this is going to be 4z to the minus 6 plus something times z to the minus 2 plus something times z to the 0 plus something. On the other hand we can take p of z cubed and multiply it by 4 and this will be 4z to the minus 6 plus something times z to the minus squared and so on. Now you see these terms cancel out and we're just left with terms in z to the minus squared and a constant term so what we can do is we can take rho prime of z squared and this will be equal to 4 rho of z cubed plus something times p of z I'm sorry I keep calling this rho I should be saying p so we take something times p to cancel out the term in z to the minus 2 and then we take some constant to cancel out the constant terms and then plus something that vanishes at z equals 0 so this is doubly periodic and it has no poles and vanishes at z equals 0 well if it's doubly periodic and has no poles it's bounded in a fundamental region so it must be constant and since it vanishes at 0 it must actually be 0 so this expression here must actually vanish so we find that bias stress p function satisfies the following differential equation plus some constant times rho which is traditionally called minus g2 for complicated historical reasons that satisfies this equation here okay well what on earth is the point of that and the other question is why these things called elliptic functions when they seem to have nothing to do with ellipses well answer the second question first because it's a little bit easier the reason why these things are called elliptic functions is if you work out the arc length of an ellipse it ends up with integrals like integral of dx over the square root of 4x cubed minus ax minus b so this is arc length of an ellipse so these integrals are called elliptic integrals now if you look at the denominator of this it looks very similar to this bit of the bias stress p function in fact you find the bias stress p function as an inverse if you sort of invert it it's you get the inverse function as the integral from something to p of the integral of dp over the square root of 4p cubed minus g2p minus g3 which should be a g2 so the inverse of this elliptic function p is more or less given by this elliptic integral and um someone i'm not quite remember who first noticed that these things are doubly periodic but the inverses of these functions are doubly periodic so doubly periodic functions end up being called elliptic functions as you can see the connection with ellipses is very round about and it's really rather dreadful terminology but it's probably too late to change it anyway now let's go back to algebraic geometry and look at the equation 4y squared equals x cubed minus g2x minus g3 so this is some sort of um cubic plane curve might look something like this so here we've got a some sort of affine plane curve and um if you look at the equation differential equation for the bias stress p function you see it looks very much like the equation for this cubic curve if we put y sorry put the four in the wrong place four should be there if we put y equals the derivative of the bias stress p function of z and x equals p of z then we see at the point x y lies on this curve so what's this saying it's saying that we've got a map from the complex numbers onto the set of onto the curve y squared equals 4x cubed minus g2x minus g3 which takes a point z to um p dash of z p of z well that's not quite true because this is actually not defined when z is in the lattice so this should really be c minus the lattice l and it maps this onto the curve well it's a bit annoying having to remove the lattice points so we can actually get a map from c to the projective curve where you remember we we turn this into a curve in the projective plane and just add a point at infinity and of course this maps the point zero to the point at infinity of this projective curve well this is a doubly periodic function so we actually get a map from the complex plane modulo the lattice l to this projective curve and it's not too difficult to check this is actually a homeomorphism of topological spaces now this projective curve is very nice if you take c and think of it as being r squared modulo a lattice that's just r modulo a lattice times r modulo a lattice so this is isomorphic to s1 cross s1 as a topological space in other words it's just a torus which you can picture like this so over the complex numbers this curve if you add a point at infinity is homeomorphic to a two dimensional torus and this implies that it can't be rational because any rational curve is the same as p1 up to a finite number of points and p1 over the complex numbers is just isomorphic to a sphere s2 so we've we've just got a a nice sphere there on the other hand the curve y squared equals 4x cubed minus g2x minus g3 over the complex numbers this is topologically isomorphic to a torus s1 times s1 and there is no way to turn a torus into a sphere by removing a finite number of points and adding a finite number of other points now if we had a rational map between these two spaces the rational map would be defined except to define that number of points and it would have an inverse that was well defined except to define that number of points so any two curves that are birational are the same topologically if you're allowed to add and add and subtract a finite number of points so these two curves are not birational um by the way I just just make a quick comment um so we've constructed um a doubly periodic function by taking a sum 1 over z minus lambda squared um except we're not retaking this because it doesn't quite converge but we can make it converge by um sort of fiddling around a little bit you might wonder why don't we do this with singly periodic functions well you can take the sum sum of 1 over z minus lambda and this does not converge well it's not absolutely convergent but it's so close to being convergent that it's very easy to make it convergent in fact you can make it convergent by summing it carefully in in um by carefully summing over lambda in order of absolute value and then it becomes conditionally convergent but not absolutely convergent and this is a periodic function it's not difficult to figure out which it's actually pi times the cotangent of pi z um when people are doing trigonometry they don't often mention this I think because you do trigonometry at um high school or something and at high school you're not allowed to talk about complex numbers um this actually gives rise to the product formula for the sine function because you can write sine pi of z is equal to z times a product over n not equal to zero of 1 minus c over n well you've got to be a bit careful about this product because that sum is not convergent and this product is actually not convergent either unless you're a little bit careful about the order um so what's the relation between these well if you take the logarithmic derivative of both sides then this identity here becomes this identity here um so um the analog of the via strice function is really just this function here incidentally this product formula also has an analog for the via strice function where that's that's getting to the very interesting theory of um theta functions but that's not strictly speaking part of algebraic geometry so I'll stop there for the moment