[MUSIC] So, Jacobi is quasiperiodic function of index one-half. And the theta function is a modular form, with respect to group S and 2 Zed. Can we write the main property of quasi-periodicity of our function? In more automorphic form, in other words, I would like to explain this additional factor in terms of Jacobi group. Quasi-periodicity. In terms, Of Jacobi group. Jacobi group In our definitions, the semi-direct product of the modular group with Heisenberg group. And we are allowed this as a parabolic subgroup of Ziegen modular group of genus two. So we consider it all Jacobi modular form as modular function on the. We can do the same with Jacobi to the series. But would put here e to the power by e M, because the index M, with Jacobi theta series, is equal to one-half. Though this is a modular form, a function, In the then, The equation of [INAUDIBLE] is equivalent to the following relation. Would act on the Jacobi theta series, with the additional third variable by a matrix, mu- lambda, 0 and we get our function with corresponding additional factor. This action, this is Sp2 action on the Siegel upper half plane. The additional vector lambda mu comes from the action on and now we have a very good explanation of this vector. This is the binary character of the Heisenberg group. More exactly, v H at. Mu minus lambda r where r is any [INAUDIBLE] parameter is equal to minus 1 lambda plus mu plus lambda times mu plus [INAUDIBLE]. I recall you that VH is the only, SL2 in the variant character. So Jacobi data series with additional Jacobi vector is a modular form with a character. With a binary character v H with respect to the Heisenberg, to the integral Heisenberg group. This is the explanation of the functional equation of Jacobi theta series in terms of the Jacobi group. Now we can study the modularity. Of Jacobi (I,Z). The first property, Is very simple, I would like to consider the matrix T, this is translation in then the function Is very simple. Tao plus 1z is equal to e to the power pi i over 4 data to data. This follows directly from the definition of Jacobi data series because n is and e to the power. Pi i n square over 4, (Tau + 1) gives us exactly this additional character additional factor. Now, I would like to analyze the modular equation S with respect to the involution [INAUDIBLE]. This property is much more complicated minus one over tau zed over tau. The transformation which we have in the series Jacobi form, is equal minus square root tau over i e to the power pi i Zed squared tau theta (tau, Zed). So, you see that as a factor, Up to a coefficient we have square root of tao. It follows then the weight of Jacobi to the series is equal to [INAUDIBLE] and I would like to prove this functional equation. We prove it, Without Poisson summation formula b I would like to use only the fact then Jacobi theta series as quasi periodic function. More exactly, we prove the following relation I would like to change that By z times tao. And we get a modular equation, M plus prime of the following. The vector in the square root is about change. Then we have e to the power pi i zed square tau, theta in tau zed tau. Now this function I would like to know by see in that was indexed now. What we have in the left hand side of this formula that is a quasi periodic function. For the latest, minus 1 over 2 Zed, plus Zed, I would like to proof then The new function c. Satisfy the similar functional equation. So I would like to proof that this new function Is again quasiperiodic with respect to the same link. If this is true, then we have two function satisfying the same functional equation. And according to the theory about the number of zero, we get then the function, the left hand side is equal to the function. The right hand side up to a constant, depending on tau. This is because, the both sides of this equation, vanishes if zed is equal to 0. So we'll have to function with the same 0 and after that we could prove that this coefficient is exactly the coefficient which we'll have in this functional equation. In order to realize this plane I have to study the function xi. [SOUND] [MUSIC]