[SOUND] The second part of this lecture. Is very, Very important for Jacobi forms in many, or in one, variable. This is the splitting principle. I would like to represent. An arbitrary weak or holomorphic or cusp Jacobi form of [INAUDIBLE] for the lattice L As a product. Of the vector valued, modular form x the vector our Jacobi theta series with characteristic with summation is taken over all classes of the geo latest module is the latest. It means that, the jacobi form I, we can represent as a scalar product of vector, valid modular tone, times the vector. Of theta serif. A H. Some remark about this formula. First of all, the order of this group, is equal to the determinate of L. So we'll have a finite sum in this summation. Jacobi Theta series with characteristic h, Is by definition the following function: E to the power pi i V plus H to the square, plus V plus HZ. [INAUDIBLE] efficient V the summation is over the For example, if L then this group has only one element, and in this case, we have the usual Jacobi theta series of the lattice L. We use this formula fore the even new modular latest E8. And also with an additional character for the Jacobi series of dm. So to prove the splitting principle we have to recognize the Fourier expansion of the Jacobian data series. We can work with weak Jacobian. Reorganization. Of Fourier expansion. First of all, I would like to write down the second function of equation for Jacobi form, for any Lam and nu and L pi into z plus lambda 2 plus mu into the power 2 pi i 1 half lambda to the square 2 plus lambda zed is equal phi tau zed. I simply put the standard factor to the lefthand side, and I modify it a little bit. This factor putting 2 here, because I would like to write down the Fourier expansion of our module of 4. So now the the 3 expansion of the function in the left hand side has a falling form. This sum, ANA, each is a problem, 2 PI N Thor plus A, that is more complicated. And we have to add the additional effector This is equal to, The summation is over n. Now I would like you to write This Fouriam expansion, Putting all we get n +lλ +1/2λ to the square. Comes, from this term. This is coefficient N tau plus zed l plus lambda. And this is equal to the function pi (tau, zed). n, l. Now we can compare the Fourier coefficient in the lefthand side and in the righthand side of this formula. So we get the following. We see the a(n,l) = a(n+l lambda) + lambda squared over 2, l + lambda. The hyperbolic of the index of the fourier coefficient on the left hand side is equal to heterobolic, no. Of this index and we prove the fallen proposition. The Fourier coefficient. A (n,l) depends only on the hyperbolic norm of its index, and on the class. N modular (l,l) is N, This is true for all the Jacobi form, form. Anamorphic or weak Jacobi form doesn't matter. So, you see that in principal we can organize a resumation in the fully expansion of Jacobi form. And now, I would like to organize it, first of all I, would like to denote by N one half of the hyperbolic non of the index of jacobi form is equal to N minus l squared/2. Then a (n,l) = a Capital N + l square over 2, l). With l, we can consider modular L. Let us fix A vector h, Modular L. Then, We get the following property. According to the last Fact about fourier coefficients or Jacobic form. We have that, we have the same fourier coefficient. If we modify h by any vector v, in l, and now I would like to write the Fourier coefficient of Jacobi form using this new index. First of all, we'll take a finite sum over all classes Of L star over L. Then, what takes the summation over all N. But N is rational in general, because the vector L is so L squared Is rational in general. So we take N congruent to minus H squared over two, modular Z. We need this condition to have an interger here. Then we take a, N plus h square over 2, h. And then, we add the summation over all vector V. In L. E 2 pi i (N+ h+v to the square over 2. So we use this fact. + (h + v, zeta)). But now, We can put these coefficient here, because it doesn't depend on v. I get here e to the power 2 pi i N. And we can cancel with N here. Then this function There's nothing else but Jacobi that the series with characteristic h. [SOUND] [MUSIC]