[SOUND] My next remark about the place of Eichler-Zagier Jacobi form in this definition. We start our course, With Jacobi form in one variable. Now we extended our main definition but this is really an extension, a generalization or a new definition. An Eichler-Zagier Jacobi form the serial of this function was constructed in the famous book of Eichler-Zagier published exactly 30 years ago. This is Jacobi form of weight K and index m. I put here Eichler-Zagier to emphasize that we have a Jacobi form in one abelian variable. We have no lattice in this definition but if we analyze the functional equation, We see then the functional equation of Jacobi form of Eichler-Zagier we had here the factor e to the power pi i pi m z to the square. What does it mean? Now I write this factor once more. So the factor in the functional equation (M), e to the power 2 pi i m the index c z to the square over c tau + d. In our functional equation with the lattice, this is e to the power pi i c z to the square, c tau d. It means that the lattice in the definition of Eichler-Zagier, Is the lattice 2 or A1. This is the lattice generated by a vector u with square 2. Means in the Jacobi form of Eichler-Zagier of weight k and index m are Jacobi form, according to our new definition, of weight k for the lattice A1 over the lattice 2 and index m. Or if you like we can take the lattice 2m and index 1. So, index M corresponds to the lattice, 2m or the lattice A1. This is the standard, the simplest root lattice renormalized by m. So this is a place of the classical Eichler-Zagier Jacobi form in our definition. Moreover, we discussed the graded ring and in Eichler-Zagier, They determined the structure of the graded ring of the weak Jacobi form for the lattice A1. They proved that this graded ring over the ring of the graded ring of modular form with respect to SL2 z has only three generators. This is Jacobi form of weight -2, 1, the principal generator we constructed in the beginning of this lecture. Then Jacobi form of weight 0 and index 1 and the search function -2, 1. These two functions we saw in the beginning of our lecture. Let me show them for you once more. This is the first function. And the second function, you can get from this function, phi- 1, 2. This is theta 2 z over eta q and tau. So you see that the theta function gives us the main generators of the graded ring of the weak Jacobi cusp form. In the case of Eichler-Zagier, the case in one variable without any lattice. This is a case of the simplest possible root system, A1. Now, I would like to construct new examples of Jacobi form in many variables and I would like to continue the series of examples of this type. So examples. Of Jacobi forms in many variables. First of all, we consider the lattice D m. I gave you the definition, but I repeat it once more. This is even sublattice The even sub lattice of the simplest Euclidean lattice of rank m. The sum of coordinate is even modular 2. Sub lattice of index 2 and z m. And for those of you who worked with the algebra and with root system, in my course I use Dm for the notation of the lattice. And then for the root system, I will use annotation R of Dm, all roots. This is all vectors of square 2 in our lattice. You can find them, for example, and you can prove without any problem, then the cardinality of the root system of Dm is equal to 2 times m(m-1). If you haven't any experience with root system, please find all rows of two vectors in this lattice. So, what kind of examples of Jacobi form for the lattice Dm, we can construct more or less for free using only Jacobi theta series. First of all, I would like to construct the first and the main generator of the graded ring of the weak Jacobi form for Dm. It was the following function of weight -m, and index 1. This is a product. Theta (z1) over eta cube (tau) and so on. Theta (z m) over eta cube (tau). This is a weak Jacobi form of weight -m for the lattice Dm. As an exercise, this is not so simple for the moment, you can try to analyse the space. For example try to analyse a dimension of the space. Using theta function we can also construct holomorphic form, for example the first function this is so-called theta series for D8, The simple product of eight function. It was our first example. We started the definition of Jacobi form in many variables from this example. This is holomorphic Jacobi form of weight 4 for the lattice D8. Later we'll see that this is a minimal possible weight for holomorphic modular form. So this weight is called the singular weight. 4 is the singular weight, means the minimal possible. For this lattice. It's really the first Jacobi form for this lattice. If we take D7, can we construct holomorphic Jacobi form for D7 using the same procedure? So if z has dimension 8, now we have 7 variables. We can do this using the same product, but we take only 7 theta series. But then the character, or it's better to say the multiplier system is not trivial. We have v added to the power 21. To kill it, we add the cube of Dedekind eta function. So we have Jacobi cusp form of weight 5 for this D7. And this is cusp form, cusp. You can continue without any problem. Please do this mathematics. D6 We take only 6 vectors, And you'll have Jacobi cusp form of weight 6 for D6 and so on. Please continue these series, moreover, now you see how you can construct Jacobi form or Jacobi cusp form for any lattice Dm. It means the product of Jacobi theta and the Dedekind eta always gives you a good example. Now, what is about another root system? Can we use Jacobi theta series to construct some rather fundamental functions for them? So for example, the first function. This is very important, because this is really the main generator of the graded ring of the weak Jacobi form. And this graded ring is very important in the theory of singularity and in the theory of forbidden structure. Now, let's analyze the lattice A n. [SOUND] [MUSIC]