Write $J_{4,6}$ for the space of Jacobi modular forms of weight 4 and index 6 for $SL_{2}(Z)$. I know that there are weak modular forms of weight 4 and index 6 for $SL_{2}(Z)$ and then I'm looking for non-weak (holomorphic or cusp) forms of weight 4 and index 6. I try to give an elementary proof that $J_{4,6} = {0}$. By elementary I want to use the valence formula for standard modular forms in one variable and the Taylor series decomposition of a Jacobi form. MY question is the Following:
Assume that the Taylor development of a Jacobi modular form F starts as follows:
$\phi(\tau,z) = \alpha \cdot \Delta(\tau)\cdot z^{8} + f_{10}(\tau)\cdot z^{10} + f_{12}(\tau)\cdot z^{12}+ f_{14}(\tau)\cdot z^{14} + ....$
or
$\phi(\tau,z) = \beta\cdot \Delta(\tau) \cdot E_{4}(\tau)\cdot z^{12} + f_{14}(\tau)\cdot z^{14} + ...$
$\Delta$ is the Ramanujan delta standard modular form in one variable on the upper half plane of weight 12. $E_{4}(\tau)$ is the standard Eisenstein modular form and $\alpha$, $\beta$ are complex constant.
My question is the Following: Can I say that the weight of $\phi$ must be at least 12 (which is the weight of $\Delta(\tau)$?
Thanks for any advise.
Thanks for your answer. I agree if you consider meromorphic Jacobi forms. I focus on holomorphic ones. Moreover I realized that $\Theta^{8}$ begins with $\alpha \cdot \Delta(\tau)\cdot z^{8}$ but has weight 4....this answers negatively to my question...However I can then build the quotient $\frac{\phi}{\Theta^{8}}$ which will be a Jacobi form of weight zero and index 2. I read in the Eichler-Zagier book (p. 11) that $dim_{C}(J_{k,m}) = 0$ for $k \le 0$ unless $k=m=0$. If it's true that solve my problem and prove that $dim_{C}(J_{4,6}) = 0$, but I doubt, it seems to be a typo in the text....Do you have an opinion?