6 F. ANDREATTA and E. Z. GOREN Then fv o L ®w(fc) = TTo Lv ®w(fc) = TT(TTo i¥ ® w(&) where for each prime ?P the last isomorphism is induced by applying OLV 8w(kp)0) to the isomorphism fv (%,i, • • •. *P,/„): W(k«p) ® W(fc)^ J J W(fc). ^ 2 = 1 For each prime ^3 and for 1 i /p, denote by e^,z eO L 0W(fc) the associated idempotent. Let p be a prime of OK over p and fix an embedding OK/P ^ k. Let J be a lifting to OKP of the absolute Frobenius on OK/P- For every prime ty of OL over p let {(TW-OL^ 0Kp}^h Jv be extensions of the homomorphisms {3"p,i} -=1 , to homomorphisms from the *P-adic completion OLV of OL to the p-adic completion OKP of O^- 2.2 Definition. Let S := Reso I / /z(G m ,o L )' Schemes • Groups be the Weil restriction of Gm?oL to Z i. e., the functor associating to a scheme S the group (r(5, Os) ®z OL) . If T is a scheme, we write ST := S x T. Spec(Z) IfT = Spec(i?), we write 2R for ST- For any scheme T define XT := HomGR(ST,Gm5T) as the group of characters of ST- We often write X for XoK. 3 Moduli spaces of abelian varieties with real multiplication. 3.1 Note carefully. Throughout this paper we fix a fractional ideal 3 with its natural positive cone 3+, among the ones chosen in 2.1. Below, we discuss Hilbert moduli spaces and Hilbert modular forms, where the polarization datum is fixed and equal to (3,3 + ). Our notation, though, does not reflect that. When we are compelled to consider the same notions with all the polarization modules chosen in 2.1 simultaneously, this will be explicitly mentioned. 3.2 Definition. Let S be a scheme. Let N he a positive integer. Denote by fm(s,tJLN)^s

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