[MUSIC] Hi, we are studying our fifth lecture and we continue the six chapter. Jacobi theta series. This is one of the main object of my course and due to this function Eichler–Zagier called the function we are talking about Jacobi modular forms. But you cannot find the Jacobi theta series in the book of Eichler–Zagier but for my course, this is really one of the main tool. By definition. This is the following series. The [INAUDIBLE] symbol, E to the power P-I N to the square over 4, tao, +nz. Summation is taken over all integers. As usual, tao are able from the upper plane and z is a complex number. 4n is equal to + or -1. If M is to + or- 1, modular 4. And zero if M is even. Let us study this function. The first question is about the convergence. This function. Converge. Absolutely. And normally. On any compact set. Of H 1 cross C. More exactly, we can make the following estimation of the general term of this function. Let's assume that imaginary part of Z is smaller than c. And imaginary part of tau which is a positive number is strictly larger than epsilon. So we estimate the general term of this series, it's absolute value. Restrict e to the power, Pi epsilon and to the square over. Which is equal to e p. N C -epsilon n 0 over 4, times e to the power -p epsilon n, n -n0 over. Now, we can fix n0 in such a way that this number is negative. Then, this is strictly smaller than E to the pi to the power nine, E to the power, n, n -n0 over. So you see that this series converges very, very rapidly. [NOISE] Now, it's easy to see that this function is anti symmetric with respect to z. Or it's better to say that this function is and z. This is directly follows from the definition because our character, the chronicle character, is in our function. Therefore. Our theta series vanishes in there. Now, I would like to study its quasi periodicity. More exactly, we prove that this function satisfies the following functional equation for any lambda in mu. In that. -1 to the power lambda + mu times e to the power -pi i, lambda to the square [INAUDIBLE] +two lambda theta, z. So you see, that here we have two factors. The second factor is similar to the factor we saw in the definition of Jacobi form. The only difference that in the definition of the Jacobi form of weight k and index m, we had the factor 2 pi i m. Therefore. The Jacobi theta series suggests phi as a correspondent elliptic equation for m = 1/2. The first one is + -1. It depends on lambda and mu. Let's prove this equation. So this is the elliptic equation of Jacobi theta series. To prove it, [COUGH] we have to use the definition of Jacobi form. This function is equal to the following sum. E to the power T i n square over 4 tau +. N z plus lambda tau +. Let's try to find a resumption in these series. First of all. M is always odd. Because they're connected character -4 over n. 4 over n is 0 4 even n. Therefore, e to the power phi i n mu is = to -1 to the power mu. We analyze this. Now, I would like to write the rest. We can write this expression in full length form. First of all, Tor times n^2 over 4 +. N lambda and I would like to add lambda to the square. Here. We get the full square. N over 2 = lambda to the square. It would denote this as (N over 2) to the square. Now. We have to cancel the additional term lambda to the square. + z(n + 2 lambda) because I would like to have here N. -2 lambda z. So we were right. The expression which we have on the exponent. So let me continue summation. So we get then our expression star (*) = to the sum over -4 over n. E to the power -p i capital N to the square over 4 tau + capital Nz -, or I can put simply here. E to the power -pi i lambda to the square 2 + 2 lambda z. Wait capital N Capital N is = +2 lambda, but then you can check -4 Over small n is equal to -1 to the power lambda the connector symbol -4 over n +2 lambda. If lambda is even this is evident. But if lambda equal one modulo four, we change one to three and three to one modulo four. So this property is more or less evident. So to finish we can change. Our character. The character -1 to the power lambda, -4 capital N. And we get. First of all, we have additional Sin -1 to the power mu, which we have to, certainly to add to this summation. And as a result, we have (-1) to the power Lambda +mu times theta series into Z times the additional vector for the. The e to the power -Ti ( lambda squared tau + 2 lambda z). Therefore, we proved that our Jacobi theta series is quasi periodic function. [MUSIC]
