# Fourier coefficients of Jacobi forms over Cayley numbers.

• Volume: 11, Issue: 1, page 125-142
• ISSN: 0213-2230

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## Abstract

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In this paper we shall compute explicitly the Fourier coefficients of the Eisenstein seriesEk,m(z,w) = 1/2 ∑(c,d)=1 (cz + d)-k ∑t∈o exp {2πim((az + b/cz +d)N(t)) + σ(t,(w/cz +d) - (cN(w)/cz + d)}which is a Jacobi form of weight k and index m defined on H1 x CC, the product of the upper half-plane and Cayley numbers over the complex field C. The coefficient of e2πi(nz + σ(t,w)) with nm &gt; N(t) has the form-2(k - 4)/Bk-4 ∏p SpHere Sp is an elementary factor which depends only on νp(m), νp(t), νp(n) and νp(nm - N(t)) = 0. Also Sp = 1 for almost all p. Indeed, one has Sp = 1 if νp(m) = νp(nm - N(t)) = 0. An explicit formula for Sp will be given in details. In particular, these Fourier coefficients are rational numbers.

## How to cite

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Eie, Min King. "Fourier coefficients of Jacobi forms over Cayley numbers.." Revista Matemática Iberoamericana 11.1 (1995): 125-142. <http://eudml.org/doc/39482>.

@article{Eie1995,
abstract = {In this paper we shall compute explicitly the Fourier coefficients of the Eisenstein seriesEk,m(z,w) = 1/2 ∑(c,d)=1 (cz + d)-k ∑t∈o exp \{2πim((az + b/cz +d)N(t)) + σ(t,(w/cz +d) - (cN(w)/cz + d)\}which is a Jacobi form of weight k and index m defined on H1 x CC, the product of the upper half-plane and Cayley numbers over the complex field C. The coefficient of e2πi(nz + σ(t,w)) with nm &gt; N(t) has the form-2(k - 4)/Bk-4 ∏p SpHere Sp is an elementary factor which depends only on νp(m), νp(t), νp(n) and νp(nm - N(t)) = 0. Also Sp = 1 for almost all p. Indeed, one has Sp = 1 if νp(m) = νp(nm - N(t)) = 0. An explicit formula for Sp will be given in details. In particular, these Fourier coefficients are rational numbers.},
author = {Eie, Min King},
journal = {Revista Matemática Iberoamericana},
keywords = {Series de Eisenstein; Series de Fourier; Coeficientes; Jacobi forms; Cayley numbers; Eisenstein series; Fourier coefficients},
language = {eng},
number = {1},
pages = {125-142},
title = {Fourier coefficients of Jacobi forms over Cayley numbers.},
url = {http://eudml.org/doc/39482},
volume = {11},
year = {1995},
}

TY - JOUR
AU - Eie, Min King
TI - Fourier coefficients of Jacobi forms over Cayley numbers.
JO - Revista Matemática Iberoamericana
PY - 1995
VL - 11
IS - 1
SP - 125
EP - 142
AB - In this paper we shall compute explicitly the Fourier coefficients of the Eisenstein seriesEk,m(z,w) = 1/2 ∑(c,d)=1 (cz + d)-k ∑t∈o exp {2πim((az + b/cz +d)N(t)) + σ(t,(w/cz +d) - (cN(w)/cz + d)}which is a Jacobi form of weight k and index m defined on H1 x CC, the product of the upper half-plane and Cayley numbers over the complex field C. The coefficient of e2πi(nz + σ(t,w)) with nm &gt; N(t) has the form-2(k - 4)/Bk-4 ∏p SpHere Sp is an elementary factor which depends only on νp(m), νp(t), νp(n) and νp(nm - N(t)) = 0. Also Sp = 1 for almost all p. Indeed, one has Sp = 1 if νp(m) = νp(nm - N(t)) = 0. An explicit formula for Sp will be given in details. In particular, these Fourier coefficients are rational numbers.
LA - eng
KW - Series de Eisenstein; Series de Fourier; Coeficientes; Jacobi forms; Cayley numbers; Eisenstein series; Fourier coefficients
UR - http://eudml.org/doc/39482
ER -

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