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Quaternion Algebras

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References for Chapter III

The Dirichlet series associated to automorphic forms on a quaternion algebra are discussed in

  1. Godement, R., Les functionsdes algebres simples, I, II Seminaire Bourbaki, 1958/1959

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  2. Shimura, G., On Dirichlet series and abelian varieties attached to automorphic forms, Ann of Math., vol 76(1962)

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  3. Tamagawa, T., On ℓ — functions of a division algebra, Ann of Math., vol 77(1963).

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They are also referred to, but rather obliguely, in

  1. Selberg, A., Discontinuous groups and harmonic analysis, Proc. Int. Cong. Math. (1962).

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The theory of Eisenstein series is discussed in [11], [13], [14], and [25] as well as in

  1. Langlands, R.P., On the functional equations satisfied by Eisenstein series, Mimeographed notes.

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  2. Langlands, R.P., Eisenstein series, in Algebraic Groups and Discontinuous Subgroups, Amer. Math. Soc. (1966)

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  3. Selberg, A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, Jour. Ind. Math. Soc, vol XX (1956).

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[25] and [28] are of course the basic references for the Selberg trace formula. Some of its formal aspects are also described in

  1. Langlands, R.P., Dimension of spaces of automorphic forms, in Algebraic Groups and Discontinuous Subgroups, Amer.Math. Soc. (1966)

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The theorem of §16 can still be stated and proved if M is replaced by a quaternion algebra which splits everywhere that M' does. The proof is in fact rather easier. However these apparently more general theorems are immediate consequences of the proof of the original theorem. Theorems very similar to that of §16 and its extensions have been proved by Shimizu. Our methods differ little from his.

  1. Shimizu, H., On discontinuous groups operating on the product of the upper half planes, Ann. of Math., vol 77(1963)

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  2. Shimizu, H., On traces of Hecke operators, Jour. Fac. Sci. Univ. Tokyo, vol. 10 (1963)

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  3. Shimizu, H., On zeta functions of quaternion algebras, Ann. of Math., vol 81(1965)

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We have also had occasion to refer to

  1. Langlands, R.P., The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups, in Algebraic Groups and Discontinuous Subgroups, Amer. Math. Soc. (1966)

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Orthogonality relations for the characters of non-compact groups first appeared in

  1. Harish—Chandra, Discrete series for semisimple Lie groups, II, Acta Math., vol. 116 (1966)

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We thank R. Lipsman for the reference to

  1. Rickert, Neil. W. Convolution of L2functions, Coll. Math v. XIX (1968)

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Jacquet, H., Langlands, R.P. (1970). Quaternion Algebras. In: Automorphic Forms on GL (2). Lecture Notes in Mathematics, vol 114. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0058991

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  • DOI: https://doi.org/10.1007/BFb0058991

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-04903-6

  • Online ISBN: 978-3-540-36234-0

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