[SOUND] [MUSIC] >> To construct the generators of the correspondent pace of Jacobi cusp form, we'll study Jacobi theta series and this is our next section, Jacobi, Theta series and examples of Jacobi modular forms. But first I recall you the property of the classical Dedekind eta function. So this section, for this section we can add another title. Jacobi theta and Dedekind, Eta. The Dedekind eta function. The Dedekind eta. Function eta is defined by the following infinite product. Q to the power 1 over 24, the product of the following factors, where n is greater or equal to one. Eta one is the well defined function on the upper half plate. And this function has the following invariant properties. First of all, this function is quasi-periodic. More exactly, adding tao plus 1 is equal to e to the power pi over 12. Theta. This is clear from the definition of Jacobi form. The next property is more complicated. Eta in minus one of it is equal square root tau b eta and tau. So this, the first equation, this a modular equation for the translation. The second equation is a modular equation with respect to the evolution. The only problem that we have the square root from tau. We fix the branch using this condition. Square root is positive if tau is positive. But these two model equation have the following corollary. Eta function is a modular form of half integral weight. Means that this functional equation is true for any m equal to a, b, c, d and s, z. Where V eta M is the root of what a 24 from 1, and this is a projective character. This means that the product. Of two values of the v eta is equal to +1 or -1, v eta, from the product of two matrices. We call this function multiplier system. This is multiplier system. Of the modular group s of 2 z. Certainly the square, of eta is the usual character. Usual multiplicative, a character of order 12 and the corresponding even power of Dedekind eta function. Will be modular form of integral weight. Okay. With respect to the full modular group, s l to zed, with a character For any k greater or equal to 1. In particular, eta function to the power of 24. This is Delta function. This is the cusp form of weight 12 with respect to the whole modular group. For this function, we have two nice identity of Euler. One can calculate the Fourier expansion of the Dedekind eta function, have the following formulas. Minus 12 over n, this so called Kronecker symbol. Q to the power N square over 24, where n is greater or equal to 1 and 12 over n is equal to 1. If n is congruent to plus or minus 1 L, minus one, if n is congruent to plus or minus pi, [INAUDIBLE] 12. And 0, if n [INAUDIBLE] are not [INAUDIBLE] This is the famous Euler formula who found also The following expression for the cube of the Dedekind eta function. Minus 4 n times n q to the power n squared over 8 n greater to 1. Where -4 over n is another chronicle symbol, this is + or- 1, if n is congruent to + or -1 model 4 or 0, if n is congruent to zero modular 2 if n is even. This is 2, formula of Euler. You can prove these formulas by yourself but certainly this is not so easy. But, some standard can be notarial argument could help you to get both formulas. Maybe in the better file, I'll give you some more detailed argument about those two formulas. But now the main hero of this section, [SOUND] Jacobi, Theta function. Jacobi theta series is a function in two complex variables. Tau is in H, z in C. By definition, this is the following function. -1 to the power n-1 over 2, E to the power pi times n squared over 4 tau plus nz, where summation is taken over all odd numbers. But we can give you another expression for the same function. This is the sum -4 over n, q to the power n square over 8, r to the power n over 2 as usual q, e to the power two P-I tau. R-E to the power two P-I-Z. When summation is taken over all that. Please compare two formulas then you can check then this expression gives us another expression for the Kronecker symbol. And what is very important and was proved by Jacobi, then We have another expression for the same function, this is famous Jacobi product formula. Then for the same function, we have an expression in terms of infinite products q minus, sorry. Minus q to the power 1 over 8, r to the power minus 1 over 2. The product of n 2 greater than 1. One minus q to the power n minus one r. One minus q to the power n r to the power -1. One minus q to the power n. So we have two different formulas for Jacobi theta series. And the properties, the modular properties of Jacobi theta series will start in the next lecture. [SOUND]
