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On adelic automorphic forms with respect to a quadratic extension

Published online by Cambridge University Press:  17 April 2009

Ze-Li Dou
Ze-Li Dou
Affiliation:
Mathematics Department, P.O. Box 298900, Texas Christian University, Fort Worth, TX 76129, United States of Americaz.dou@tcu.edu
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Abstract

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Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 1 , August 2000 , pp. 29 - 43
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Dou, Z.-L., ‘On a theta correspondence with respect to a quadratic extension’, J. Number Theory 79 (1999), 1–18.CrossRefGoogle Scholar
[2]Dou, Z.-L., ‘Theta correspondence and Hecke operators relative to a quadratic extension’, Acta Arith. 93 (2000), 237–255.CrossRefGoogle Scholar
[3]Dou, Z.-L., ‘Arithmeticity and theta correspondence on an orthogonal group’, in Proc. New York Number Theory Seminar, (Chudnovsky, , Chudnovsky, , and Nathanson, , Editors) (Springer-Verlag, Berlin, Heidelberg, New York) (to appear)Google Scholar
[4]Shimura, G., ‘On certain zeta functions attached to two Hilbert modular forms, II’, Ann. Math. 114 (1981), 569–607.CrossRefGoogle Scholar
[5]Shimura, G., ‘On the critical values of certain Dirichlet series and the periods of automorphic forms’, Invent. Math. 94 (1988), 245–305.CrossRefGoogle Scholar
[6]Shimura, G., ‘On the fundamental periods of automorphic forms of arithmetic type’, Invent. Math. 102 (1990), 399–428.CrossRefGoogle Scholar
[7]Shimura, G., ‘The critical values of certain Dirichlet series attached to Hilbert modular forms’, Duke Math. J. 63 (1991), 557–613.CrossRefGoogle Scholar
[8]Yoshida, H., ‘On a conjecture of Shimura concerning periods of Hilbert modular forms’, Amer. J. Math. 117 (1995), 1019–1038.CrossRefGoogle Scholar