Revista Matemática Iberoamericana

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Volume 11, Issue 1, 1995, pp. 125–142
DOI: 10.4171/RMI/168

Published online: 1995-04-30

Fourier coefficients of Jacobi forms over Cayley numbers

Minking Eie[1]

(1) National Chung Cheng University, Chia-Yi, Taiwan

In this paper, we shall compute explicitly the Fourier coefficients of the Eisenstein series $$E_{k,m}(z,w) = \frac{1}{2} \sum \limits_{(c,d)=1} (cz + d)^{–k} \sum \limits_{t \in o} \mathrm {exp} \{ 2\pi im (\frac{az+b}{cz+d} N{t} + \sigma (t, \frac{w}{cz+d}) – \frac{cN(w)}{cz+d}) \}$$ 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 $$– \frac{2(k–4)}{B_{k–4}} \Pi_p S_p .$$ 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.

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Eie Minking: Fourier coefficients of Jacobi forms over Cayley numbers. Rev. Mat. Iberoam. 11 (1995), 125-142. doi: 10.4171/RMI/168