Fourier coefficients of Jacobi forms over Cayley numbers

Abstract

In this paper, we shall compute explicitly the Fourier coefficients of the Eisenstein series

which is a Jacobi form of weight $k$ and index $m$ defined on $\mathcal{H}\times {\mathcal{C}}_{\mathbb{C}}$, the product of the upper half-plane and Cayley numbers over the complex field $\mathbb{C}$. The coefficient of $e^{2\pi i(nz+\sigma (t,w))}$ with $nm>N(t)$, has the form

Here $S_p$ is an elementary factor which depends only on ${\nu }_p(m)$, ${\nu }_p(t)$, ${\nu }_p(n)$ and ${\nu }_p(nm–N(t))$. Also $S_p=1$ for almost all $p$. Indeed, one has $S_p=1$ if ${\nu }_p(m)={\nu }_p(nm–N(t))=0$. An explicit formula for $S_p$ will be given in details. In particular, these Fourier coefficients are rational numbers.