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Abstract

We prove an algebraicity result concerning special values at critical points, in the sense of Deligne, of tensor product $L$-functions associated to automorphic representations of special orthogonal groups for quadratic forms which are totally definite, and, cuspidal representations of ${\rm GL}(2)$ corresponding to primitive cusp forms, over totally real number fields. We also prove the reciprocity law, i.e., the equivariance under the action of ${\rm Gal}({\overline{\Bbb Q}}/{\Bbb Q})$, for the special values. In the appendix, the second author calculates the Deligne periods for such $L$-functions, assuming the existence of corresponding motives and the automorphic transfer to a quasi-split form of the special orthogonal group. Our result conforms with the celebrated conjecture of Deligne on special values of motivic $L$-functions.

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