On the critical values of \(L\)-functions of \(GL(2)\) and \(GL(2)\times GL(2)\)
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.
scientific article; zbMATH DE number 624865
scientific article