INTRODUCTION 5

For a fixed maximal lattice L in V we have a natural order S in the algebra

A(V) containing L. For each nonarchimedean prime v we put Jv = S* (1 G+(V)V

and Yv = Sv Pi GJr{V)v, where the subscript v indicates localization at v as usual.

We fix an open subgroup J of

G

+

( V ) A

whose nonarchimedean factor is of the

form Y[v Jv with an open compact subgroup Jv of G+(V)V such that Jv — Jv for

almost all v. We let q denote the set of all such v. We then let y denote the set

of elements y G

G

+

( V ) A

such that yv eYv if v G q and yv G Jv otherwise.

We can consider a formal Hecke algebra 9t(j7\ y) consisting of formal finite

Q-linear combinations of double cosets JaJ with a G y; we can similarly de-

fine 9t(Jy, Yv). We now consider a C-valued function f on

G+(V)A

such that

f(caxw) = ip(c)~1f(x) for every a G G + (V), every w e J whose archimedean

component is 1, and every c G F£, where ip is a fixed Hecke character of F. As-

suming that f is a Hecke eigenform, that is, f is an eigenfunction of every element

of 91(^7, y), we have a homomorphism Xv : 9t( Jy, 1^) — C for each f G q defined

by f| JvaJv = Av(J-uaJv)f for every a eYv.

To define our Euler product, we first consider a local Dirichlet series whose

"denominator" gives the Euler v-factor, which can be given by

(6) EV(\TTV\S) = ] T Xy(JvaJy) . Wa)\s, A = JV\YV/JV,

aEA

where s G C and | | is the normalized valuation in Fv. We can determine (Theorem

8.24) the explicit form of (6) as a rational expression in |7rv|s. NOW, our Euler

product, twisted by a Hecke character \ °f E, is of the form

[n/2]

(7) z(S, f , x)=n L^2S+2~2i

^x2)

n

MXMKI8),

2=2 vEq.

where Lq(s,

V'X2)

is the function obtained from the Hecke L-function of

ipx2

by re-

moving the ^-factors for all v £ q. The product Yu=2 ^q eliminates the numerator

of \\vec.Ev. Before proceeding further, let us insert here a remark. If dim(V) = 5,

we can choose (p so that G+(V) is the group of similitudes of an alternating matrix

of size 4, and

G1(F)

= Sp(2, F). Then the series of (6) is the local version of the

series introduced in [S63].

Let us recall here the Euler product on O^. Fixing v, put Bv = {7 G 0% | Lv~f —

Lv}. Then we can consider the Hecke algebra 9t(J3v, 0%); given a Hecke eigenform

g on O^, we can define a homomorphism /JLV : 9\(BV, 0$) — C. Then the Euler

^-factor is given in the form

(8) Ev(\7rv\s) = £ iiv(Bv£Bv) • 7V(dil(0)"S, X = Bv\0*/Bv,

where dil(£) is the denominator ideal of £, and iV(dil(£)) is its norm. Notice that

£ in (8) can be any element of the group 0£, while a in (6) is restricted to "integral

elements" belonging to Yv. In [S97] we gave an explicit rational expression for (8).

Now the Euler product for g twisted by \ is of the form

[n/2 (9) z(s, g,x ) =n]

L^2S+2

- ^

x2)n KixMM8)-

2 = 1 v G q