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A note on fields of definition

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A note on fields of definition COMPOSITIO MATHEMATICA M. KAREL Anote on fields of definition Compositio Mathematica, tome 45, no 1 (1982), p. 109-113. <http://www.numdam.org/item?id=CM_1982__45_1_109_0> © Foundation Compositio Mathematica, 1982, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 109 A NOTE ON FIELDS OF DEFINITION M. Karel* COMPOSITIO MATHEMATICA, Vol. 45, Fasc. 1, 1981, pag. 109-113 © 1982 Martinus Nijhoff Publishers - The Haque Printed in the Netherlands Introduction The purpose of this note is to show that the space of linear combinations of certain Eisenstein series on a rational tube domain is defined over an explicitly computable ground field in the following sense: the space has a basis of automorphic forms all of whose Fourier coefficients lie in the ground field. We also take this oppor- tunity to correct an omission in an earlier version of this theorem, which appeared as 5.5.3 in [3]. To illustrate and motivate the theorem, consider for a positive square-free integer m the normalizer T°(m)+ in PSL(2, 1R)0 of Hecke’s congruence subgroup To(m); see [1]. It is known that T0(m)+ is a maximal discrete subgroup of PSL(2, R)0 and that it has just one cusp. In particular, it therefore has one Eisenstein series of each weight satisfying the condition that the first Fourier coefficient be 1. An immediate consequence of what we are going to prove is that all the Fourier coefficients of these Eisenstein series are rational numbers. The earlier version of our theorem is not applic

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