[MUSIC] The next root system of the next lattice is the lattice An. An, what is it? This is the standard hyperplane. In Z n + 1. More exactly, These all integral vectors. In Z n+1, the sum of coordinates of the vector is equal to 0. And this is not only a sublattice of the Euclidian lattice Z n+1. But this is a sublattice of D n+1 because the sum of the coordinate is even. The root system, Of this latest, this is the root, System, An. So, I repeat that I use the letter A, D, E for the latest, not for the corresponding root system. Please calculate the number of two vectors. And this lattice. Now how to construct a Jacobi form. First of all, I would like to construct the principal weak Jacoby form for these lattice. And I can use the fact that the root lattice An is a sublattice of D n+1. We can construct the following function. So for the latest D n+1, we take this quotient, eta / theta. But for the last factor, we can tell more. We have the expression for the last coordinate in terms of the previous coordinate. So here we have to put theta in (-(z1+ zn)) / theta cube. So now let us calculate what function do we get. So we have a function, t of weight- (N + 1) for the root lattice An and this is a weak Jacobi form. So this is a weak Jacobi form, Of weight -(n + 1), An. And this is really the main generator of the corresponding gradient ring of weak Jacobi form. I don't like to go into detail, but here certainly we have to consider symmetric Jacobi form in their end with respect to the permutation for A. So as you see in any case for using only Jacobi theta series, we can construct lots of interesting function. Using the same principle, like for the latest Dm, you can construct very many Jacobi forms. For example, to have a Jacobi form without character, we'll have eight factors. And now I can construct a jacobi form for the latest A7. How to construct amorphic Jacobi form. Start with the product, Z7, seven factors, times, The theta in the sub-frame of the Jacobi form of weight 4 for A7. So you see then this is really function related ways theta series for D8. A simple exercise which I would like to formulate in the text. This function, Phi4 A7. This function is a cast form or not. Please analyze the Fourier expansion of this Jacobi form. So using the same principle, I would like to repeat using the same principle. You can construct many Jacobi form for the root system Am. The next example, very important example for our consideration. I would like to consider, that even unimodular lattice E8. Unimodular means that this lattice coincides with its dual. And for this lattice, For the following Jacobi form, we discuss this function briefly in the introduction in the first lecture of this course by definition. The Jacobi thetaseries for this lattice is equal to the sum e to the power Pi i (the scale is vector v tau the sum / ov and D8 + 2v Z. It's possible to prove that this is a Jacobi form of weight 4 for the latest E 8. I don't like to prove this fact now, but there's been an equation. This is simple exercise. And to prove the modular function equation, you have to use the Poisson summation formula, E8 is unimodular, so the Poisson summation formula gives us exactly the correct functional equation so some technical question. And I hope that you can really do it by yourself or I try to prepare it in the PDF file related to this lecture. So like you see that the idea of Jacobi theta series is really a generalization of this idea. We have a generalization of usual theta series, remind you of the definition usual theta series in one modular we're able. This modular form of weight form with respect to S into that group. So really, the similar argument, related to the Poisson summation formula, gives us the functional equation for the Jacobi theta series in 8, additional variables. Do have any kind of relation between these function and the original classical theta series. Because for theta series certainly you know than the usual theta series, Is the Eisenstein series. It's identical to the Eisenstein series, and for Eisenstein series, we can calculate its Fourier expansion that gives us a lot of very nice classical arithmetic formula. What do I have in the case of Jacobi form? First of all, using this function, Gives, this function gives us many examples, Of the classical Eichler-Zagier Jacobi form. Let me explain you this construction. With x, A vector u in E8 of square to m is not 0. Now I would like to take the following vector z = U times z where z is a complex number. We consider the following function. This is by definition the pullback Of the Jacobi theta series for E8, for that equals to uz. And using the functional equation, was a Jacobi theta series in 8 variable. We get that this function is a Jacobi form of weight 4 and index m where m, this is the square of vector u / 2. This is very nice function and I can calculate its fully expansion. The sum of a pole b and e 8 e to the power pi i, e to the squared, tau +2 vuz. This is the definition of this pullback. And now there's the following Fourier expansion. We start by one, plus This is sum of n in z positive, l in z, a(n,l). e to the power 2 pi i (n tau + l z), where a(n, l), a(n,l) is a number of 4 vector to v in E8 besides v to the square = m. And the, this scalar product of vu = n. So as we discussed in the first lecture, the original function gives us the distribution of our vector controls the distribution. And its pullback gives us very nice generated function related with the vector of ea. [MUSIC]
