[SOUND] Our next sub title is Jacobi. Modular, or cusp forms of critical aid. According to the theorem of there are no Jacobi form of critical weight. In the case of but, we constructed a Jacobi form of critical weight in our course. Let's take a vector, v, In the latest D8. And we analyzed the pull back of Jacobi form for D8 on the orthogonal complement of the vector v. This is a product of 8 Jacobi tethered series data zed 1 to data zed 7 times the 8 factor minus zed 1 plus zed 2. And so on to zed 7, this is Jacobi form, of weight 4 for the latest, A7, because the latest A7 is by definition. The orthogonal complement, it was a vector V and D8 4 is the critical weight. For A7, I put you a question, this is a cusp form or not? Please try to answer but now I would like to give you an example of Jacobi cusp form of critical weight and we know, we're sure that this is a cusp form. So now the first example, Of cusp form. We're taking as a vector u = 2(1, and 2). This is a primitive vector in the lattice D8. This vector is primitive because the sum of the coordinate without the first factor 2 is odd. Then, according to the serum proved in the last lectures, the pullback of Jacobi theta function, theta D8 on the orthogonal complement. Of this vector u is a modular form Jacobi form of this with respect to this lattice. Moreover, we can calculate the order of this function at infinity that remind you since the order of Jacobi form, it infinity. This is the minimal values of hyperbolic. Nomes two M minus eight to two squared. Taking overall non-zero for your coefficient and we'll prove than the order on this pool back is great or equal to one over u to the square. And this is equal to 1 over 4 times 7 plus 4, 1 over 44, this is positive. Therefore, this is Jacobi casp form of weight 4, with respect to this lattice. The rank of this lattice is equal to 7, so generally this function is not 0. Identically, because we know all 0s of the Jacobi so you see, if we have more than one variable. Then it might happen that we can construct Jacobi cusp form of critical weight. So this interests in question to define this dimension and in principle we can put this question now by the dimension formula. So is the next sub-title, dimension formula for J K, L. From our splitting principle, immediately we'll have that the dimension of this space is equal to the dimension of the vector valued modular four ofrade K minus N zero over two cotol. With respect to the complex conjugation of the right representation or maybe I write simply here as representation complex conjugation. And like for the case of modular form in one variable, you can calculate this function. You can find this formula in the recent Freitag's paper. [SOUND] Formula [SOUND] so I'll give you the explicit reference. It was this Frei tag's paper, and the bit there fine, and rate it to the scores but I would like to emphasize that. Usually, Frei Tag's use the method of use the formula to calculate the Dimension Formula for the vector of the modular form. This formula is very explicit and very useful, but for some small Weight, like critical or singular. Weight, we certainly have some difficulties with application of Riemann Roch Formula. Moreover, what weights are small for us from many points of view. Let me fix the least of small weights for the lattice l of rank n0. The minimal possible weight are certainly n0 over 2, this is singular of the minimal possible. Then, we'll have the critical weight. Then we can go to the weight in zero and this is so called Canonical. Sorry, as is this not N 0 I am sorry but this is M 0 plus 2. This is canonical. Now I explain it, why this weight is canonical. For example, I assume that n 0, understand this, n0 is equal to 1. Then using Jacobi form of weight k and index m, we can construct, using my lift in construction. In modular form on the plane, H2. Moreover, let me repeat this, so Phi is Jacobi cusp form of weight 3 and index m. So, 3 is exactly here, 1 + 2 is equal n0 + 2, then. Some constructions, the lift construction of this function gives us, a Siegel Modular form, a cusp form of weight 3, with respect to so called paramodular group, Gm. Where Gm is a subgroup of the group of the group operational point of simplistic group of genus 2. [SOUND] And in particular, if denote this function is phi, in the F index phi. Then the following function is that, is that a standard differential on the signal for uplane is canonical differential form on Sigel trifled. This is why this weight is called canonical. Moreover, now I am very sketchy, without any proof, it follows from this construction than if the space of Jacobi cusp form, Isn't real, then H3,0 of this, modular three fault. Is non trivial or in the [INAUDIBLE] of this para modular group is non [INAUDIBLE]. So you see then the dimension formula for Jacobi cast form plus some construction of modular form for larger group. Which we can get using Jacobi form, are very, very useful in mathematics. So coming back to the dimensional formula. You can analyze this paper. The exact reference I'll give and to analyze what type of Jacobi form you can really construct Using this dimensional formula. But to finish this lecture, I would like to analyze the component of the modular form, so our problem now to analyze. The combatant fh tau as modular forms with respect to congruence. So we need one more formula. [SOUND] [MUSIC]
