PREFACE A preliminary idea of writing the present book was formed when I gave the Prank J. Hahn lectures at Yale University in March, 1992. The title of the lectures was "Differential operators, nearly holomorphic functions, and arithmetic." By "arith- metic" I meant the arithmeticity of the critical values of certain zeta functions, and I talked on the results I had on GL2 and GL2 x GL2. At that time the American Mathematical Society wrote me that they were interested in publishing my lectures in book form, but I thought that it would be desirable to discuss similar problems for symplectic groups of higher degree. Though I had satisfactory theories of differ- ential operators and nearly holomorphic functions applicable to higher-dimensional cases, our knowledge of zeta functions on such groups was fragmentary and, at any rate, was not sufficient for discussing their critical values. Therefore I spent the next few years developing a reasonably complete theory, or rather, a theory ade- quate enough for stating general results of arithmeticity that cover the cases of all congruence subgroups of a symplectic group over an arbitrary totally real number field, including the case of half-integral weight. On the other hand, I had been interested in arithmeticity problems on unitary groups for many years, and in fact had investigated some Eisenstein series on them. Therefore I thought that a book including the unitary case would be more appealing, and I took up that case as a principal topic of my NSF-CBMS lectures at the Texas Christian University in May, 1996. The expanded version of the lectures was eventually published by the AMS as "Euler products and Eisenstein series." After this work, I felt that the time was ripe for bringing the original idea to fruition, which I am now attempting to do in this volume. To a large extent the present book may be viewed as a companion to the previous one just mentioned, and our arithmeticity concerns that of the Euler products and Eisenstein series discussed in it I did not include the cases of GL2 and GL2 x GL2. Those cases are relatively well understood, and it is my wish to present something new. Though the arithmeticity in that sense is the main new feature, as will be explained in detail in the Introduction, I have also included some basic material concerning arithmeticity of modular forms in general, and also a treatment of analytic properties of zeta functions and Eisenstein series on symplectic groups which were not discussed in the previous book. It is a pleasure for me to express my thanks to Haruzo Hida, who read the manuscript and contributed many useful suggestions. Princeton February, 2000 Goro Shimura Vll
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