-
On special values of certain
L -functions, II - American Journal of Mathematics
- Johns Hopkins University Press
- Volume 138, Number 4, August 2016
- pp. 1117-1166
- 10.1353/ajm.2016.0037
- Article
- Additional Information
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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.