[SOUND] [MUSIC] What is the simplest vector in the lattice E8, the simplest nonzero vector? This is a root. This is a vector with the minimal possible positive square, 2. In this case, the Jacobi theta series. E8r. Start 1 + q something is a Eichler–Zagier Jacobi form of weight 4 and index r to the square over 2y. We see that the pull back of this function, its value for z equal to 0, this modular form of weight 4 with respect to [INAUDIBLE] z. But there is only one modular form of weight 4. There is only one up to the constant. But the first Fourier coefficient, 0, 0 Fourier coefficient of the theta series is 1. So this function is identical. To the Eisenstein series of weight 4. And we saw this Eisenstein series in our lecture today. So we found a Jacobi form of weight 4 at the index 1. And its pullback, its particular value for z equal to zero, is the Eisenstein series of weight 4. It means that we proved the following proposition about Eichler–Zagier Jacobi form. Proposition. The space of Eichler–Zagier Jacobi form of weight 4 and index 1 is generated by Jacobi theta series. And we see it doesn't matter what root you would take. I maybe remind you that the number of roots of this lattice is equal 240. Let us prove this proposition. If you take an arbitrary Jacobi form of weight 4 and index 1 in one variable, then its value for z equal to 0 is up to the constant, the Eisenstein series of weight 4. Therefore, the function phi prime, which is equal to phi(tau,z) minus c theta E A r. Is equal to 0 for z equal to 0. Now, using the Taylor expansion of this function. This function is an even weight. So we only have only even powers of the variable z in the Taylor expansion. The first coefficient is a cusp form of weight 6 for S equal to z, but this space is trivial. I will remind you that we'll prove that if our function starts from the positive power of z, meaning if its value for z equal to 0 is equal to 0, then the first nonvanishing Taylor coefficient is the cusp form. So, you see, then the order of 0 of this function of weight 4 and index 1 is greater or equal to 4. But according to the theorem about the number of zeros of Jacobi form, we know that Jacobi form of index m has exactly 2m zeros in any fundamental domain. But here we have four zeros. Then this function is identically zero. The theorem is proved. The proposition is proved. So the pull back of the Jacobi theta series gives us very important functions. Moreover, like in the case of the usual modular form, when the theta series is the Eisenstein series, we can prove, and this result was proved in the book of Eichler–Zagier, then this function. Is the Jacobi Eisenstein series of weight 4 and index 1. And its Fourier expansion. Is found in the book of Eichler–Zagier. So analyzing the Fourier expansion of this function, we can find really the explicit arithmetic formula for this very important arithmetic function. In particular, as I mentioned in the first lecture, you can find the explicit number of representation of natural numbers by the lattice E7. The fact that we discussed in this part is rather general. Let me generalize our consideration. Let's consider an arbitrary holomorphic Jacobi form of weight k for a lattice l. Then the corresponding pullbacks gives us Eichler–Zagier modular form. By definition, this function is a pullback of the Jacobi form phi in many variables for z equal to uz. And this is holomorphic Jacobi form of weight k and index the square of u over 2 Eichler–Zagier. Certainly, the modular equation, two modular equation are evident in this case. But now I would like to analyze the condition to be holomorphic in infinity. So why this function is holomorphic Jacobi form. Let's analyze the Fourier expansion of this function. The Fourier expansion of the original Jacobi form in many variables has the following form. 2n minus l,l to the square is greater or equal to 0, where l is in the dual lattice. Now, we have to change. Instead of z, we take the vector u z. So instead of this scalar product, we get. Z times l,u. And for the Jacobi form of Eichler-Zagier type. We have the following condition. So 4 times n by the index. Minus the square of the second index should be greater or equal to 0. Why this is true? So let us rewrite this formula. We get 2 times n(u,u) minus (l,u) to the square. But l, the scalar product, to the square is less or equal l to the square times u to the square. This is a standard inequality, because the lattice, I would like to remind you that the lattice l is positive definite. So we have that this is greater or equal 2n(u,u) minus (l,l)(u,u). This is equal u to the square times 2n minus l to the square. And this is greater or equal to zero. So we proved then this function. Is holomorphic. At infinity, in the sense of Eichler–Zagier definition. But why I made this calculation. You see that we could have the strict inequality in this case. If this is true, then we have a stronger relation. It means then sometimes the Eichler–Zagier pullback of Jacobi form of many variable may be cusp form. So let me repeat this fact, and we'll start. And I really give an illustration of this effect in the beginning of the next lecture. So maybe this function is not holomorphic. We construct a lot of examples of this function, for example, the Jacobi theta series for E8. So we start with non holomorphic function. And we could get a cusp form here. Theoretically, this is possible due to this condition. And this is really the case. And in this way, I can give you a new explanation and really good theoretical explanation of the construction of theta quarks. This is the most interesting construction which we had in the first part of our course, but we give the elementary proof why the theta quarks is holomorphic Jacobi form. But in the next lecture, we give the first conceptual proof. We really explain then that theta quarks very nice cusp form, we can get very naturally using Jacobi forms in many variables. [MUSIC]