This result is very important. So we prove, then in the free expansion of our pullback. And the scalar product of l and u is always nonzero. Let us analyze the general idea of the pullback. We see, that in this particular case, in the case of Jacobi, theta series for Dm. The Fourier coefficient, au, nlM, gives us Fourier coefficient, is not equal to 0. Then automatically, 2n minus lm lm is greater or equal lu to the squared but we. Now, this lu in our case. Solve the vector, lu. Now K, this is lu, the scalar product U over u, u. It follows add u to the square is always greater or equal. You get this is not zero, 2 u over u, u. To the square and equal to one over U, U. Now I can formulate this result as a theorem. [SOUND] With proof, the falling theorem. If u is equal to two times b one b m in D, M, and u is primitive. Then, The, Order, at infinity the pull back. Of the Jacobi of the series of Dm on your octagonal is greater or equal to 1 over u, u. In particular it mean that this function will be cast form. Maybe with a character, maybe Jacobi way, but it will be cast form. The hyperbolic law of all known zero Fourier coefficients will be positive. Let's consider an application. Take the vector u, go to 2 times 1, 1, 1 in this way. Then this vector is primitive. And its orthogonal complement in this array is zero lattice A2, according to our definition. Now let's calculate this, Pullback. We get theta and z1 times theta z2 times theta minus z1 plus z1, And the order of this function, At infinity, Is greater than or equal to 1/12 according to our theory. But the order at infinity of the [INAUDIBLE] of the series. This 2 times 1 over 24, according to our definition. This is 1 over 12, it follows then the following function. Theta z1, theta z2, theta minus z1 plus z2 over eta, let's note this function is theta A2. Is holomorphic, Jacobi form of J1 for the latest A2. The only difference with the standard definition is the following. Here we get the character, or the type v theta total power 8. The character of 43 because theta function has multiply system V8 a cube. So, we have three theta function and one eta function. Though this result, Is very nice because starting from. The toroidal theta product, we get a Ramanujan theta function. In principle, this function is very well known in the theory of [INAUDIBLE] algebra. So this is the eliminator function of the algebra type A2. But now, we can construct theta block. As a pull back of this new function. [SOUND] Now, we can construct theta quarks. We take the vector AB minus AB anoted by V in A2 then the theta-quarks, Constructed in the lecture, 6. [SOUND] Is a restriction a power data function for A2? Though this is a conceptual explanation, y theta quarks r morphic Jacobi form as an exercise, I put you the following question. We gave an answer on this question in the lecture six, but now try to use a new argument related with this construction. And try to find all M and B. Such that, z theta, is what we call the Jacobi form. If you don't like the character of what is real in this construction. Certainly, you can take the product of 3. [SOUND] 3 data function for A2. We can construct, The following theta function. For three direct copies of A2, By definition, this is direct of the tensor product of 3 different functions, Theta A2 and Z1, Z2 theta A2. And z3, z4 and theta A2 in z5, z6. This is a Jacobi form, Of weight 3. Plus 3A2 with [INAUDIBLE] character. 3A2, I mean the direct octagonal sum of three [INAUDIBLE] of A2. I would like to emphasize then we get this [INAUDIBLE] non trivial function using, >> [NOISE] >> A very simple product of three, Jacobi theta series. And the construction, >> [NOISE] >> Of pullback on the sublattice of 1. [SOUND] I think I will explain why the classical Jacobi of the series theta tau. That is very very useful function in our consideration. But now we would like to develop the general theory of Jacobi theta functions in many variables, and you will see that some constructions simpler in more clay in this general case, than in the case of in one complex break. [MUSIC]