INTRODUCTIO N
The firs t mai n them e o f thi s boo k i s to associat e a n Eule r produc t t o a n auto -
morphic for m tha t i s a Heck e eigenfor m o n a classica l grou p i n a suitabl e sense .
As the nam e indicates , Heck e treated th e cas e of holomorphic modula r form s wit h
respect t o a congruenc e subgrou p r o f 51/2 (Z). Eac h Heck e operator i s given b y a
double cose t Far wit h a belongin g t o a semigrou p E o f matrices , containin g r,
whose entrie s ar e integer s an d determinant s ar e positive . Takin g a n eigenfor m /
in th e sens e tha t f\rar A (a)f wit h a comple x numbe r A (a) fo r ever y a £ E,
one consider s a Dirichle t serie s o f the for m
(1) !(*) = £ A(a)det(a)-* .
aer\s/r
With a suitabl e choic e o f E on e ca n sho w tha t X ha s a n Eule r produc t whos e
Euler factor s hav e degree 2; moreover F(s)1(s) ca n be continued t o a meromorphi c
function o n th e whol e s-plan e wit h a t mos t tw o simpl e poles ; F(s)%(s) i s entire if
/ i s a cus p form .
Various kind s o f generalizatio n o f thi s theor y o f Heck e hav e bee n attempte d
and carrie d out , bu t an y attemp t mus t b e precede d b y a choic e o f formulatio n
that i s practicable . Thoug h on e migh t b e abl e t o presen t a framewor k includin g
every imaginabl e Eule r product , i t i s o f littl e us e i f on e canno t indee d prov e th e
desired results. Therefor e i n this book we choose one type of Euler product whic h is
somewhat differen t fro m Hecke' s type. T o be explicit, w e take an algebraic numbe r
field K wit h an automorphism p o f order 1 or 2; we then put F =
{x€K\xp
= x}
and
(2) G(p) = G* = {ae GL n(K) \ ap
lap
= p }
with a fixe d ip G GLn(K) suc h tha t
t (pp
etp, e ±1 . Thus ou r grou p i s eithe r
symplectic, orthogonal , o r unitary . Suppos e tha t w e can spea k o f a n automorphi c
form wit h respec t t o a congruence subgrou p F o f G^ o n a space on which G^ acts .
Taking a certai n subgrou p E o f G^ whic h contain s F an d i s dens e i n almos t al l
nonarchimedean localization s o f G^ an d a n eigenfor m / o f al l
FOLF
wit h a G E,
we consider a serie s
(3) *(*) = X ) A («M"ra
aer\~/r
with a suitabl e functio n v o n G^ an d X(a) define d a s abov e i n th e presen t case .
Strictly speakin g on e ha s t o formulat e everythin g o n th e adelizatio n G ^ o f G^,
and th e serie s o f (3 ) i s the righ t on e onl y i f "th e clas s numbe r i s one" i n a certai n
X l l l
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