[MUSIC] Hi there, participants. So today, we'll start our lecture number six. We're nearly in the middle of our course, and today we can start to do mathematics. First of all, let me recall you what we have considered, what has been done in our course. First of all, after a short introduction about generating and partition function, we introduced Jacobi modular forms as usual modular forms with parameter. So Jacobi form, Jacobi modular forms. The first definition, we gave using two relations. Modular relation, which shows that Jacobi form is a modular form, with respect to the first variable. And the so-called elliptic equation, which shows, then our function is a billion function with respect to the variable theta. Then, we gave another interpretation of this object. We consider it Jacobi form as a function on the plane, where the third variable is rather formal, when added with index M, this coefficient M is exactly the index of Jacobi form. And we considered this function as gamma g modular form on z upper half plane H2. Gamma J in this context, Jacobi group. Is amorphic to the semi-direct product of the usual modular group, SL2(Z) times the circle Heisenberg group. And we consider it, this Jacobi group, as a parabolic sub-group of group, of genus two. Using this interpretation, for example, we can reformulate the definition of Jacobian modular form to be holomorphic at infinity. Because we had two functional equation, the modular equation and the elliptic equation, plus some condition to be holomorphic at infinity. But this third condition is equivalent to the fact that our Jacobi form consider it as a modular form on H2, is simply holomorphic on H2 In particular, from this we immediately get the fact that all Jacobi forms form a gradient ring because the product of two holomorphic functions is holomorphic. So the second interpretation of our modular group, a subgroup Sp2Z, is very useful. Especially when we consider the Jacobi theta series. So we defined Jacobi theta series. And we proved then the Jacobi theta series with the additional factor, E to the power pi i, orbing here. So, you see that corresponding index m here is equal to one-half. Then, this modular form, Jacobi theta series considered on the upper-half plane is a modular form of weight one-half and index one-half with the following multiplier system. This is the multiplier system of the function to the power of 3 times, the binary character of the Heisenberg group. VH with this the binary character of the Heisenberg group. The explicit formula for it is given by -1 to the power mu + lambda + mu times lambda + R. Exactly this factor, mu times lambda, this term, we don't see if we consider Jacobi theta series as a function of two variables, tau and zed. Because these additional terms Cams from the variable m omega. So this is a binary character, and this character, this is very important, is invariant with respect to the conjugation by any element from the module group S. So, and now I would like to put the following question. In what sense we will do mathematics today. I would like to pose the following question or problem. To construct Jacobi forms using only two blocks. The first block is Dedekind eta function. The second block is the Jacobi theta series. And to construct as many Jacobi modular form as possible. This is a problem which I would like to consider today. I would like to add then you cannot find the Jacobi theta series in the book of idea Jacobi modular forms and really, the subject of this book we discuss in our course. But this interpretation of Jacobi modular form as a function on the Ziegel upper half-plane was used very much in my joint paper with Viacheslav Nikulin. Please see the exact reference in the PDF file. This is a paper about automorphic forms and [INAUDIBLE] algebra. So this is a paper, from '96, '98, and now, I would like to show you this technique to put some new questions. Moreover, now we can construct Jacobi forms which you cannot find in any books or papers. We're doing mathematics. Now I would like to fix the type of blocks which we, which we'll consider today. First of all, we proved in the last lecture the following fact. If we multiply Jacobi theta series by, if we multiply the second variable by a, we get a Jacobi form of weight one-half and index a to the square over 2 with respect to the following multiplier system, the cube was [INAUDIBLE] function times the power a of the Heisenberg character. So, we proved this fact in the last lecture and I would like to use the following notation. I denote this function simply as theta of the index a. So certainly we can use this function as a standard block, especially because if A is even, the Heisenberg character will be [INAUDIBLE]. So the most general function which we consider at the moment has the following form. Add the function the power d times Theta a1, Theta a2, so on, theta ak where ae is strictly positive. This is not a restriction because Jacobi theta series are old functions. But certainly, if we put here a equal to 0, the product will be 0, because we know that Jacobi theta series as the only divisor. This is an integral point point of the latest Z Tau + Z. So and we would like to get a Jacobi modular form with trivial character. Let us control the character of this product. First of all, the multiplier system has the form d + 3 times k. Assume that this is equal to 1, identically. Moreover, I can calculate the Heisenberg part of the character. So VH to the power a1 + so on + ak which is also equal to. So the first condition gives us d + 3k = 0 module 24. In particular, d is 3e Is divisible by 3. And the second condition gives us the following restriction. The sum of the powers is even. Now, I would like to consider all product with this condition. [MUSIC]
