[SOUND] [MUSIC] Hi there colleagues, we are starting the lecture 10 Our current title, auto morphic direction of Jacobi form, This construction Works very well in the case of Jacobi forms in one variable, the case of [INAUDIBLE] Jacobi forms. But also we can use it for Jacobi forms in many variables but let me start was the following theorem. You can find it in the paragraph of the book. On Jacobi forms, the formulation of this theorem is the following. Let phi be a weak Jacobi form of weight, k, and index m so we consider this innovation described in [INAUDIBLE]. Then, we can consider Its Taylor expansion in Zed, for Zed = 0. And all theta coefficient, coefficient FF tau are quasi modular form N plus K, I give you today The different proof of this fact, our method, I call this method automorphic correction. Works very well in this case, it gives different proof of this result of all the vectors I give. Moreover, we can generalize this method without any problems to the case of Jacobi forms in many variables. And in the case of many variables this result is our form of automorphic correction gives very nice and very constructive result. Which is better than this formulation in the case of Jacobi forms in operator. But first of all, I would like to discuss Quasi-modular forms, what are they? And our first sub section today will be modular differential operator. We start. With a differential operator. In tao with this normalization 1 over 2 pi i. We use this normalization because we can derive this differential operator in terms of our formal variable q, so q, this as usual, 2 to the power of 2 pi i, tau. And this operator, act on any holomorphic or miramorphic function. In the following we'll have to multiply four coefficients by its index error. This is why I used this normalization. So z differential operator transform holomorphic form into holomorphic but this operator is not modular in general so it means then if I denote this by d then d doesn't keep. The property to be modular but in one special case, this is still true. Let's consider the action of this operator on Jacobi modular function of weight 0. So we can consider weak modular form. Weak means that we accept forms at infinity. In the Fourier expansion of this function has the following form. The index n in greater equal to minus N, where N is a natural number. So, this function has a pull of order capital M infinity and now I would like to analyze the modular behavior of the action of the differential operator on f, maybe I can use other slide. F is a modular function. It means that this function is invariant with respect to all element in the who ordered group. Now we can apply at the rate, D to the left and to the right to the side and of the support it. On the left hand side we get C tau plus D to the power minus 2, D F zero A tau plus B over C tau plus D is equal to d of 0 and tao. So the differential operator d transforms weak modular functions modular function was infinity. Into the space of weak modular form of weight 2, with respect for modular. So for the weight 0, the is a differential operator, D is a module operator. What can we do, For the classical modular form of weight k, I would like to consider the following operator. So we fix. A modular form of weight K. Then. We can define the following. We divide the modular form f by The [INAUDIBLE] function to the power 2k then we apply the differential operator d. And then, we'll multiply the result by the same power of the data function. I would like to analyze this operator and to calculate what function you will have. The element for calculation shows that it's equal to df times eta to the power of 2k Minus f 2k eta to the power of 2k minus 1 D eta and we have to divide by the power, not 2, by the power of 4k, the square of the denominator. The result Is the following (Df) tau- 2k, then we have the logarithmic derivative of the function times Is a modular form f. So this is our DK. What type of modular transformation do we have for this function? The theta function is modular, with respect to the full modular group. Therefore, this function. Is a modular form. Of weight 2. Weak modular form maybe with some infinity with respect to the full modular group, S h2 zed. But with a character, the character will be the multiplier system of the Dedekind function to the power -2k. This is a second factor but we have to multiply this function to the first factor. So we see that this product in fact belong to the space of maybe weak modular form of weight 2, of weight 2 + k with To have a better formula for this function, I have to analyze the logarithmic derivative of the data function. And this is [SOUND] our next task, so I would like to calculate, [INAUDIBLE] derivative, of the decay data function, where, The function is given by the following infinite product. We have used this product several times in our course. Our problem now is to calculate the logarithmic derivative of the Is that maybe I write down that it's modular property. This is a modular form with weight one-half, with respect to full modular group Zed, to Zed with multiply system, where v eta is the root for the 24 from 1. [SOUND] [MUSIC]
