JournalsrmiVol. 11, No. 1pp. 125–142

Fourier coefficients of Jacobi forms over Cayley numbers

  • Minking Eie

    National Chung Cheng University, Chia-Yi, Taiwan
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Abstract

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

Ek,m(z,w)=12(c,d)=1(cz+d)ktoexp{2πim(az+bcz+dNt+σ(t,wcz+d)cN(w)cz+d)}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 kk and index mm defined on H×CC\mathcal H \times \mathcal C_\mathbb C, the product of the upper half-plane and Cayley numbers over the complex field C\mathbb C. The coefficient of e2πi(nz+σ(t,w))e^{2 \pi i(nz+\sigma (t,w))} with nm>N(t)nm > N(t), has the form

2(k4)Bk4ΠpSp.– \frac{2(k–4)}{B_{k–4}} \Pi_p S_p .

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

Cite this article

Minking Eie, Fourier coefficients of Jacobi forms over Cayley numbers. Rev. Mat. Iberoam. 11 (1995), no. 1, pp. 125–142

DOI 10.4171/RMI/168