[MUSIC] So I would like to analyze the last property of the Jacobi theta series. I would like to explain why Jacobi theta series is holomorphic as Jacobi modular form. With this, we have to analyze [SOUND] the Fourier expansion of Jacobi theta series. The Fourier expansion, we have in the definition of Jacobi theta series. The symbol q to the power n square over 8 r to the power n over 2, n is in z. Let me write down the Fourier coefficient in explicit form. The Fourier coefficient depends on (n squared / 8, n / 2) and this Fourier coefficient is equal (-4 / n), this is +/- 1 or 0. But now we can calculate the hyperbolic norm of the index of this Fourier coefficient of the Jacobi modular form of weight 1/2 and index 1/2. So as a norm of the index is equal by definition 4n m- l to the square where m is the index. In our case, this is equal to 4 times n square/8 times 1/2- (n/2) to the square, but this is 0. See, very nice property, and if our Fourier coefficient Is not 0, it follows that it's hyperbolic null is 0. Certainly, the Jacobi form is holomorphic, as a Jacobi form holomorphic at infinity. But if you would like to analyze usual modular form with the same property, what we get if we write down Fourier expansion of usual modular form, then this property will have only for the Fourier coefficient a(0) to the constant term. So, from this point of view, the Jacobi theta series generates a property of constant 1. But you see that the Jacobi theta series has a lot of non-zero Fourier coefficient but the hyperbolic null of all indexes is equal to 0. So now I would like to analyze the first example. So why we studied in all details the Jacobi theta series? Because using this series, we can construct a lot of examples of Jacobi modular forms. First of all I can construct a very simple weak Jacobi form with triple character. Will put by the definition theta function to the square over the 6 power of the theta function. So the multiplied system of this function is triple. The character of the Heisenberg group to the square is also triple. So we get a Jacobi form of weight -2 and of index 1 because Jacobi theta series has index 1/2. So, this function is very nice. In principle, we could construct a function, with a character. Jacobi theta series over the cube negative function. The weight is -1, the index is 1/2, and the character is only reduced to the character of the Heisenberg group, the binary character. We can find the first Fourier coefficient for the function of weight -1. The Fourier expansion starts with (r to the power 1/2- r to the power -1/2) + q times something. Please check this. Fourier square, we get an (r-2 + r to the power -1) + q(). So this two Jacobi forms are very very important because we can prove that the space of weak Jacobi form of weight -2 in index 1 is generated by this function. I can give two proof of this simple fact. First of all let us analyze the Taylor expansion. The constant term is the modular form of weight -2. Certainly we have no holomorphic form of weight -2. It's a constant term of 0. And we know that zed equals 0 is the divisor of this function. Then we have a modular form of weight 0. And this a modular form of weight 0, is respect an S to that, therefore, this is a constant. And if this constant is equal to 0, the correspondent. So here, let me change the notation. We can analyze an arbitrary, Weak Jacobi form of weight -2 and index 1 then it's constant term certainly 0. The second Taylor coefficient is a modular form of weight 0 with respect to the full modular group. Therefore, it's a constant. But if this constant is 0, Then the function psi has more than than two divisors, more than two 0s. Because the order will be at least 4, so this function is identically 0. But the order 0 of our function phi -1 is exactly 2. So let me write down the divisor phi -1 = 2(0) modular, zed(2) + Z. We found all weak Jacobi form of weight -2 and index 1. We analyzed all weak Jacobi forms of weight -2 and index 1, but now I can construct the first cusp form. This base of Jacobi cusp form of weight 10 and index 1 is generated by one function, phi 10, 1, phi 10, 1 in zed. This is dedicated to the function to the power. 18 tau times the square [SOUND] of the Jacobi theta series. First of all you see that this function phi 10, 1 is delta function times phi -2, 1. I write it once more to the square. So this is explicit formula for this function and you see then the index Of theta squared is equal to 1, the weight is 1 and the characters (V to the power 6). This is a character, not multiplied system but really character, of order 4 of the full modular group. So to kill this character, we add the theta additional vector. And certainly we get a cusp form because tethered to the square is holomorphic because the product of two holormorphic in Jacobi form of holomorphic but theta function contains some power of q. So the product will be cusp form, why? We have only one cusp form of this type. Let's take another three cusp form of weight 10 and index. Let us analyse the Taylor expansion. The first coefficient is the cusp form of weight 10. Certainly this is 0. Then the second coefficient, according to the proposition we proved in the last lecture is a cusp form of weight 12. And up to a constant there is only one cusp form of this weight. So if this function, this coefficient 0, then psi identically 0 because this function will have a 0 or at least four in 0 that equal 0. Then this function is identically zero because the index is one. So, we see that the space of Jacobi cusp form for 8, 10 and index 1 is one dimension. [MUSIC]