AUTOMORPHIC FORMS ON INNER FORMS OF GL(2)
This chapter is an exposition of the ideas and classical (and modern) results of the theory for Hilbert modular forms and quaternionic automorphic forms. It starts with a brief exposition of the theory of quaternion algebras. After a short description of functorial algebraic geometry of Grothendieck, quaternionic automorphic forms (including Hilbert modular forms) are introduced in adelic terminology. An elementary description of the theta correspondence and the Jacquet-Langlands correspondence between quaternionic automorphic forms and Hilbert modular forms follows. Finally, the integral solution of the basis problem (of Eichler) is presented.
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