[SOUND] [MUSIC] Now we can formulate the main theorem of the first part of our lecture. This is a theorem about the modular correction of the heat differential of the rate. That's 5b in Jacobi form of weight k for the latest N. Then the following operate, H, index K, which is equal to H plus (2k- n0) quasimodular differential operator and quasimodular [INAUDIBLE] series. G2 in tau transforms Jacobi modular forms of weight K for lattice l into the space of Jacobi modular form of weight K+2, for the same lattice l. Certainly, the similar result we have, For the space of Jacobi cast form. And the image of a cast form is a cast form and the similar fact, we have for the space of Jacobi weak form. I'm sorry, I forgot to add that we change the weight. The image has weight K plus 2. Please compare this result. With. The differential operator Dk, which transforms the space of modular form of weight k Into of module 4 of K+2. And it was a modular differential operator DK, our formula has a form. The differential operator D + 2k times [INAUDIBLE]. So, you see that for the modular form, the [INAUDIBLE] modular form, I have a very similar module correction. The only difference, then we have to take into account the rank of the latest L over [INAUDIBLE] variables of [INAUDIBLE]. Now, I would like to discuss the proof of this [INAUDIBLE] a form is a function which satisfies two functional equations. The first equation was a modular group S A 2 Z. The second is for the Heisenberg group, or the second transformation is the fact that our modular form was quasi periodic with some factor, with respect to all translation. Of the variable z. Now, what do we have in the first case for the modular translation? We proved then the heat operator is module operator in the case of the single away. In that we have to consider the falling transformation. We have to apply the heat of curator in Jokovia form of single await and zero class and 0/2. To get this Jacobi form, we have to divide. Jacobi form phi by eta to the power 2k, we get a Jacobi form for 0 -n0. This gives us the Jacobi form of singular weight. And then, to get the original weight, we have to multiply by the same factor. This is very similar to the procedure we have used when you can strike them with module correction of the usual differential of the rate in case of module form in one [COUGH] anyone on it. But, now if I make a similar co-relation we'll get here exactly separator HK. H plus two K minus N zero G two it, so we have this coefficient. Pick up into denominator We have (2k- n0), so this is our operator HK. So we prove, then, H K commutes or it's better to say than A H K [SOUND] P such as five. SA2 function equation will weight K plus 2, just the heat operator change the weight. So it was starting from the k from the singular weight, we'll get the singular weight plus 2. [SOUND] So H sub k respect is a modular action, with respect a special in group SA2, but HK change the way what is about the modular transformation. Now we're coming back to the definition of the differential curator. We define our heat H, as a data operator for the hyperbolic lattice L1. And [SOUND] for any latest M this differential operator, certainly is in [INAUDIBLE]. [SOUND] With respect to the full orthogonal group of the real space. Complex space doesn't matter, of the latest M but our Jacobi forms and our differential operator, will get from this construction. This is action On the hyperbolic differential operator on this function. And the abelian transformation, the transformation with respect to z, let me write it here, if we take b The translation of that, with respect tob The vector lander. Tower for lander, this average for vector. So this transformation we can, can write down one, going to write it. In terms of our nominal group, O U plus A minus 1, so it mean that, our definition of the hit operator as differential operator are related to the hyperbolic lattice. Gives us the invariance with respect to this transformation more or less automatically. So this is in the definition of our differential operator of the second order. Then variance with respect to mu if at here + mu is more or less evident. So I hope that you understand that my general arguments, so we proved [SOUND] that the function H K pie. Transforms like Jacobi form of weight K plus two. So but why this function [SOUND] is Jacobi form. So now I would like to prove that H K [INAUDIBLE] is holomorphic. At infinity. For that we have to calculated the Fourier expansion of Jacobi form. If we take the original Fourier expansion function phi. Then HK 5 into z is equal to one half the action of the hit operator gives us the additional coefficient to any coefficient. In particular, we cancel all coefficient with hyperbolic zero of its index. Then plus 2K minus n zero G2(tau) times phi(tau, z) but G2(tau) has the following Fourier expansion. We found this Fourier expansion In the previous lecture. And now we see that after multiplication, we have only Fourier coefficients, which satisfy our standard condition for the hyperbolic. Moreover, if our Jacobi form Is a cast form, then H K F is cast. The same is true for the weak form because we get no fourier coefficient with negative. Power of tower, so our serum is completely proof. [SOUND] [MUSIC]