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On the critical values of $$L$$-functions of $$GL(2)$$ and $$GL(2)\times GL(2)$$. (English) Zbl 0838.11036
Let $$\lambda$$ be a primitive cuspidal automorphic form for $$GL_2 (F)$$, where $$F$$ is a number field. According to well-known conjectures (Shimura, Taniyama, Langlands), there should exist a rank 2 motive $$M (\lambda)/F$$ attached to $$\lambda$$ such that the $$L$$-function $$L(M (\lambda) \otimes M (\varphi), s)$$ is equal to the Hecke-Langlands $$L$$-function $$L (\lambda \otimes \varphi, s)$$ for all arithmetic Hecke characters $$\varphi$$ of $$F$$, where $$M (\varphi)$$ is the rank 1 motive attached to $$\varphi$$ by Deligne. Moreover, if $$M (\lambda) \otimes M (\varphi)$$ is critical, then according to Deligne the special value $$L(M (\lambda) \otimes M (\varphi), 0)$$ divided by a certain period should be algebraic. Although in the general setting it seems difficult to compute these periods, one can predict how they change up to algebraic values when one varies $$\varphi$$; in fact, their ratios should be given by simple determinant factors coming from the comparison isomorphism between the Betti and de Rham realizations of the $$M (\varphi)$$’s.
In the present paper, the author under certain conditions — without supposing the above conjectures about $$M (\lambda) \otimes M (\varphi)$$ — shows the existence of certain quantities attached to $$\lambda$$ which behave just like the motivic periods described above. Similar results are proved in the context of Rankin $$L$$-functions attached to two automorphic forms $$\lambda$$ and $$\mu$$ and a Hecke character $$\varphi$$. In the case where $$F$$ is totally real, results of a similar type as proved here have been given previously by several authors, including Manin, Mazur, Harris and Shimura.

##### MSC:
 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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