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
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