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
( 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
( 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
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,
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
(7) z(S, f , x)=n L^2S+2~2i
2=2 vEq.
where Lq(s,
is the function obtained from the Hecke L-function of
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
= 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]
- ^
x2)n KixMM8)-
2 = 1 v G q
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