Notes on Jacquet-Langlands Theory（英文版）

　　The purpose of these notes would have been better explained if we had chosen another title， namely， Jacquet-Langlands' Theory Made Easy； it occurred to us at the last moment that a more pedestrian choice would be more prudent， since after all the author is in a rather bad position to judge.These notes cover a very large part of Chapters 2， 3， 5， 6， 9， 10 and 11 of Jacquet-Langlands' work， Automorphic Forms on GL（2）， VII 548 pp.， 1970， Springer （Lecture Notes in Mathematics， No. 114）. Since the volume of our notes is about one fifth of 548 pp.， it is not to be expected that we have been able here to explain everything. In fact， we have entirely omitted the explicit construction of discrete series from quadratic extensions or quaternion algebra （Chapter 4 of Jacquet and Langlands）， the connection with Zeta functions of matrix algebras （Chapter 13）， and the most interesting， or at any rate newest， part of their work， namely， the relations between the "spectra" of a quaternion algebra and a 2×2 matrix algebra. The reader who is sufficiently interested by the present notes will of course have to go back to Jacquet and Langlands anyway.We have given full proofs in Chapter 1 and nearly complete ones in Chapter 3， but not in Chapter 2. For the bibliography， we refer the reader to Jacquet and Langlands， where references will be found.These notes have been written after lectures on the same subject at The Institute for Advanced Study， where we found， from September， 1969 to April， 1970， a very welcome atmosphere of quiet intellectual work. It is for us a great pleasure to express here our deep gratitude not only for the conveniences we were provided with， but also for the fact that we were spared the duty to thank the U.S. Air Force for its main contribution to Culture and Civilization， namely， the highly palatable Napalm-and-Mathematics cocktail that is the mark of the times in the most advanced country of the world.

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