PREFACE A substantial portio n o f this volume i s based o n my lecture s a t th e NSF-CBM S Regional Researc h Conferenc e hel d a t th e Texa s Christia n University , Ma y 19-24 , 1996. Whe n I was asked to give lectures at the conference and to write up eventually the content s o f the lecture s i n monograph form , m y idea was rather differen t from , if no t unrelate d to , wha t I a m presentin g now . A t tha t tim e I though t I woul d include th e result s I ha d publishe d i n a serie s o f papers , whic h concerne d Eule r products an d Eisenstei n serie s on symplecti c an d metaplecti c groups , an d I woul d also discuss th e arithmeticit y problem s o f the specia l value s of the Eule r products . After thinkin g abou t thi s projec t fo r a fe w weeks , I foun d th e ide a unexciting . Though th e questio n o f arithmeticit y ha d neve r bee n full y explore d i n thos e case s and I stil l inten d t o trea t i t o n a futur e occasion , th e whol e progra m lacke d th e allure o f making m e brave th e burde n o f writing a book o f fair length . Therefor e I decided t o take up something ne w and mor e challenging which had bee n occupyin g my min d fo r som e time, an d o n which I had onl y incomplete result s bu t fel t tha t I had enoug h technica l idea s to complete them . However , i n additio n t o th e obviou s question o f whether thos e idea s wer e enough , ther e wa s anothe r problem , namely , whether th e propose d boo k coul d b e accessibl e t o man y readers . Afte r a fe w mor e months o f experimenting , I convince d mysel f tha t I woul d b e abl e t o accomplis h my aim s satisfactorily , an d bega n th e wor k o f whic h th e outcom e i s th e presen t volume. What ar e the n th e mai n feature s o f the book ? Leavin g th e detail s t o th e Intro - duction, le t u s merel y sa y tha t ther e ar e thre e chie f objectives : (i ) th e determina - tion of local Euler factor s o n classical groups, in an explicit rationa l form (ii ) Eule r products an d Eisenstei n serie s o n a unitar y grou p o f an arbitrar y signature (iii ) a class number formul a fo r a totally definit e hermitia n form . Though thes e for m th e principa l ne w result s obtaine d i n th e book , w e star t with quit e a genera l setting , an d includ e man y topic s o f expository natur e s o tha t the boo k ca n b e viewe d a s a n introductio n t o th e theor y o f automorphi c form s o f several variables. W e eventually specialize our exposition to unitary groups , but w e treat them as a model case so that th e reader can easily formulate th e correspondin g facts in other cases. Fo r that purpos e we find unitary groups better than symplecti c groups a s will be explaine d i n the Introduction . Princeton, October, 199 6 Gor o Shimur a

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