[MUSIC] Good afternoon, dear colleagues. Lecture eight. The subject of our discussion today. The first subject is theta-quarks. As pullbacks. Of Jacobi modular forms. In many variables. We constructed theta-quarks two lectures ago, and they are very interesting, Jacobi form in one variable. Jacobi forms of Eichler–Zagier type of weight 1 and index a2 + ab + b2 to the square with a character of order 3. Using these theta-quarks, we constructed for example, the first cusp form of weight 3 and index 13. In the last lecture, we introduced Jacobi modular form in many variables. A Jacobi form. Of Jacobi form of weight k and index a has the following free expansion. Z now is an element of the complex vector space of dimension n0 where the rank. Of a is equal to n0. So we have the following Fourier expansion a, n, a, n is not negative. A in the dual lattice a star equals the power 2pi (n tau + the scalar product of (n,d)). This is a Jacobi form. Weight k with respect to the lattice n. This modular form, Jacobi modular form, is holomorphic, it's infinity. If we have non-zero Fourier coefficient only for indices n, a, satisfying the following condition. In the last lecture, we gave the following interpretation of this condition. If we consider the index, Of these Fourier coefficient as a vector in the hyperbolic lattice U + l(0) negative, l negative different. Then this condition, Means that our vector. Vector f for example, is inside the positive cone of this hyperbolic lattice. Or on the boundary of this cone. If the hyperbolic noam of this vector is equal to 0, moreover we, Consider the following example. I remind you the definition of Jacobi theta series of the lattice. So your modular lattice E8, today we consider more general function. The following function, the summation is taken of a whole vector in the even modular lattice. E8 is a Jacobi form of weight for the lattice E8. Please analyze the definition because today, we consider more general functions. In particular, we have the following correlation between, Jacobi form with many variable, variables and the Jacobi forms of Eicher-Zagier type. We'll have a map from the space of Jacobi form of weigh k to the lattice a to the space of Eichler-Zagier Jacobi form in one variable with the index u to the square / 2 for any vector u(l). L is even integral, therefore u to the square = 2m. And in this map we get Jacobi modular forms of Eichler-Zagier type of index m. How we construct this map, for arbitrary, Jacobi modular form. In many variables, we can construct its restriction. On the line, zu = phi (tau, u times z) where z is a complex number. In particular, it works very well if our vector is equal to 0. In this case we simply take If u = 0, then we get phi (tau,0), which is the modular form of weight k with respect to the full modular group. Now, I would like to consider the restriction, or the pull back, of Jacobi form phi on some sublattice more exactly on a sublattice of corank 1. We take now nonzero vector u in n and we take its orthogonal complement in the lattice l. We get a sublattice in l. Moreover their diagonal sum of the sublattice m and the one-dimensional lattice generated by u is a sub lattice a finite index in L. L is sublattice in the dual lattice in the dual lattice of N. Sublattice of the direct sum of the dual lattices of M and U. The dual lattice of U Is generated by one vector, U over the scalar product of U. If I take now that in the complex vector space, By L then we can write down this vector is the sum orthogonal sum of two components ZM and small z times u, Where small z is a complex. The pullback which I would like to analyze Is zeros friction. Apology Jacobi form. For z = 0. We denote this function by phi in this u tau times ZM, and this is clean. This is in Jacobi form of the same weight K for the lattice M. I remind you that use it M is a. M is orthogonal content of U. What type of re-expansion has this function? To analyze the Fourier expansion, I would like to write down in habit through vector L. In the dual lattice as a sum of two vectors, lm+ lprime. Now I can write the expected formula for this vector. So let's analyze this question more carefully. lis a vector in the dual lattice which is sublattice of M* + Z times u / (scalar square of u). So then l = LN + orthogonal projection of L on this one-dimensional sublattice. So we get (l, u) times u / (u, m). And I would like to denote this vector as l and there's u. Now we are ready to write down the Fourier expansion of, the Jacobi form phi, consider it at the point small z = 0. [MUSIC]