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
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.
February, 2000 Goro Shimura