[MUSIC] So we proved the theorem about theta quarks. Let me write down this formulation once more. So the theta-quarks are products of three theta functions divided by one eta function. So this is a theta block times of types three theta, means the product of zeta over one eta, and you see in the beginning of our lecture we put this question, how many Jacobi forms we can construct using only theta and eta. But why we analyze only product? We did it in the first part of this lecture. But at the end we find this construction. We can see they're not only products, but also quotients. And this is not the only possible quotient. This is theta block of type theta. Theta ab. The product of three theta blocks of this type, the product of three theta quarks, gives us Jacobi form of weight three. Can we generalize this construction? Yes, this is really possible. For example, I can give you an explicit formula for the Jacobi form of weight 2, and index 37. This is the first cusp form of weight two. See please the book of Eichler and Zagier about Jacobi modular form. In this book is the author found the first several Fourier coefficient of this Jacobi form. But now I can give you the explicit formula for this function. Theta 5 times theta 4 times theta 3 to the squared times theta 2 to the cube times theta to the cube over eta to the power of 6. So here we have 10 theta over 6 eta. Certainly, this is Jacobi form of weight two and some index, and this is holomorphic Jacobi form. And in our case, in our particular case, this is a cusp form. So now we can calculate as many Fourier coefficients as we like, because using for example you can get all Fourier or possible Fourier coefficients of this function. So you see that this idea of theta block is very, very fruitful. So, this is my formula from 2002. But now, certainly, we can give you better explanation why theta-quark, and why these theta blocks of times 10 over 6 are holomorphic Jacobi forms. So we, at the moment, have at least three proofs of this fact is our joined project with Nils Skuroppa and Zegier, which I hope will finish maybe the next year. But, I can give you one more result, maybe it's the best possible and very useful theta blocks which we have at the moment. This is the theta blocks of type 34 theta over 30 eta. Certainly, this, again, in Jacobi form of weight 2 and some index. But you see, then the structure of this theta block is much more complicated. At least in the block, you'll see in my joint paper with Cris Poor and David Yuen. [SOUND] Maybe this result from we found these examples two years ago. And I cannot tell you that this subject is totally finished, because in the moment we don't have the full classification of theta block. We have some algorithms, how to check that this theta blocks are holomorphic modular form, but we don't have the full theory of them and you see that it might be that all Jacobi forms we can get using some constructions of this type. For weight two, you see, J was finished by J2 because according to the classical result of Nils Skoruppa, this is his, the theorem, very important theorem from his PhD thesis from '95. I think he, no '95, '85. I'm sorry. He proved that there is no, Nils-Peter Skoruppa, then there are no Jacobi form of weight one. All the spaces are free of them. So, the way two is minimal possible, and at the moment I cannot tell you when we really know the structure of Jacobi form two, in terms of theta blocks. But this object you can continue, and now I propose you to develop this subject using not only this type of theta eta but [SOUND] another type. So one more example. Let us consider the following function, which I denote by theta 3 over 2, Theta times theta in 2 Zed over theta Zed. This is the holomorphic Jacobi form of weight one half, index 4 minus 1/2, 3/2 with multiplied system v eta times VH. And to check then this is holomorphic Jacobi form, I can propose you to prove that we have the following Fourier expansion for this function. Now have connect a symbol of level 12 Q. This is the same character which we use in the Fourier expansion of the [INAUDIBLE] function or [INAUDIBLE] Kronecker symbol from the Euler formula. Here we have n to the square over 24 times r to the power n over 2. Please prove this Fourier expansion. And you can do it using the Fourier expansion for theta series and for the function. And now you see that in the theory of theta blocks, we can add also theta quotient because we still have holomorphic Jacobi form. Now I can put the following question for you. So I started this lecture by the following table ,so please construct a similar table for the theta products with theta three half. Then you have to consider this relation not modular 8, but modular 24, so try to generalize, to generalize This table to the product with theta three half. Certainly the table will be much longer, but you can formulate very many interesting questions. So today, we're doing mathematics. [SOUND] [MUSIC]