[SOUND] [MUSIC] I can formulate a similar equation for the root lattice A and this two root lattice we studied carefully in the previous lectures. So the series of formulas is here, this is the serum about what root lattice. This is the serum about root lattice A1 and after o, a you get 2m. So it means that this theorem, the theorem about the structure of the graded rink of Jacobi form for the root lattice A1. [SOUND] A1 case. What do we have for? [SOUND] The case, An. What function can we construct? We can construct the function tai minus n plus one an by definition this is a product of the following theta series. Theta in that one, over eta to the power of 3 so on till (zn). And then z1 so on till zn over eta tau cubed. We can construct the product of such in our examples but now, we have a weak Jacobi form. This is symmetric, weak Jacobi form of weight- (n + 1) for the latest An. And again, you can use differential operator to construct more generators because if we consider A1. Then we get here theta zed one times theta that one. This is kai a riser. Tridle case, when A1- this is in some sense, we cannot use this construction for N equal 1 in terms of the lattice z n but you see like a limit case you get it. If I apply the differential operator. With a root lattice An, you get a Jacobi form of weight -N+1 for An with symmetric. And my question is to find, to calculate the q0 fourier coefficient of 5 minus N plus 1, An and 5 minus N plus 1 An. So you'll see that the differential is a module of differential of generator. It really help, to help us to solve some problem about the generators of the greater three of weak Jacobi form. Now, I would like to give you some introduction in the subject of some lost like so, we consider now Jacobi type form in many variables. These functions appeared in our lecture, when we consider correction of Jacobi form. A Jacobi type form I has three variables. Tau, calligraphic Zed and Zed and, it satisfies the full and functional equation. Let me write this equation and then we discuss all. So we have an [INAUDIBLE] and the usual complex ad. And then, our function satisfies the usual function equation of a Jacobi form of weight k With respect to a ladies, L, And it has index M, was respect to the secant over able small z. And here is the usually label tao, set and set tai is the upper half plain as usual. Calligraphic z in the complexification of the lattice, and small z is a complex number. N is non-negative integer, it maybe then M is equal to 0. We accept this, more over, the zen of A. Is no negative, it means we accept [INAUDIBLE] lattice a because the [INAUDIBLE] lattice L is graphic is zero and we would have no able of this type. This is the first of the modular equation. A, b, c, d in Sl2 that. This is the first equation. The second equation is the usual modular abeline equation but, only with respect to the variable, Zed telegraphics. This is our usual modular grace for any lambda and mu in L. Therefore, in the variable small that in this variable, we have only modular equation with respect to the group S L to D. The space of this function will denote by T g Jacobi type form of weight K, K was respectfully lattice L, and in index M with respect, with assert, with variable Z. Index M search only is Rosa formal because, if we consider, for example, a Jacobi form in this space, Then, After this trigger modification of, the variable Z, in the case if m is not equal to 0. We get Jacobi type form of weight k, with the lattice L and index 1, it means that in fact there are only two spaces, principally different spaces. Jacobi type form of weight K, index L and index 1 -- N Jacobi type form of weight k w with the latest l at index 0. If this m is equal to 0, If m is equal to 0, we simply Do not have that in the modular equation. So, you see that this is really invariant function and we can have no problem to construct such Jacobi form, Jacobi type form. So how we can construct a Jacobi type form of weight of index 0? We can take formal Taylor series, Phi, K plus n and Tau and Ref to be said, Z to the power and is greater than zero, where Z coefficient Jacobi form of weight K +n for the latest l. Moreover, we have a transformation between two spaces. It means that zetamorphic correction e to the power minus 8 p to the square G2 tau Z to the square. The multiplication by the separate. Cancelled, The index one. We would like to consider this type of relation in the next lecture. [SOUND] [MUSIC]