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HK] M. Harris, S. Kudla, On a conjecture of Jacquet, in H. Hida, D. Ramakrishnan, F. , Contributions to automorphic forms, geometry, and number theory (volume in honor of J. Shalika), 355-371 (2004) [HKS] M. Harris, S. Kudla, W. J. Sweet, Theta dichotomy for unitary groups, JAMS, 9 (1996) 941-1004. [HKS] M. Harris, S. Zucker, Boundary Cohomology of Shimura Varieties, III, M´emoires Soc. Math. France, 85 (2001). [Ho] R. , 880 (1980) 211-248. [I1] A. Ichino, A regularized Siegel-Weil formula for unitary groups, Math.

1]. For general W , we can take Kh to be the stabilizer of the diagonal hermitian form diag(a1 , . . , an ) where ai ∈ Q, ai > 0, 1 ≤ i ≤ r, ai < 0, r + 1 ≤ i ≤ n. The exact choice of hermitian form has no bearing on rationality, though it may be relevant to integrality questions. With respect to the canonical embedding of Shimura data (G(U (W ) × U (−W )), Xr,s × Xs,r ) → (H, Xn,n ) we may assume KH ⊃ (Kh × Kh ) ∩ H. 1)], we may assume that the point (h, h) maps to hH . This antiholomorphic map is the restriction to Xr,s of a K-rational isomorphism ∼ φr,s : RK/Q (Xr,s ) −→ RK/Q (Xs,r ).

W] A. , 113 (1965) 1-87.

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