Jacquet-Langlands��Փ ��(n��i)�ݺ���

����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.