\(L\)-functions and periods of polarized regular motives.

*(English)*Zbl 0859.11032This is the first of several articles whose goal is to prove Deligne’s conjecture on critical values of \(L\)-functions for a rather large class of motives over \(\mathbb{Q}\). In the first part of this article we study regular motives – absolute Hodge motives with Hodge types of multiplicity \(\leq 1\) – with coefficients in a totally real number field. When such a motive is polarized – endowed with a bilinear form \(B:M \otimes M\to\mathbb{Q} (-w)\) in each of its realizations – then the Deligne periods of its twists by motives attached to Hecke characters of imaginary quadratic fields factor as products of square norms, with respect to the polarization, of arithmetic vectors of pure Hodge type. This makes computation of Deligne periods practical for motives attached to automorphic forms.

The second part of the article describes properties (partly conjectural) of motives associated to self-dual cohomological cuspidal automorphic representations of the adeles of \(\text{GL}(n, \mathbb{Q})\). These motives, realized in the cohomology of Shimura varieties attached to certain unitary groups, are both regular and polarized, hence the theory of the first part applies. The third part of the article uses the Piatetski-Shapiro-Rallis integral representation of the standard \(L\)-function of unitary groups, and techniques developed by Shimura and the author, to compute the critical values of these \(L\)-functions in terms of period integrals. The conclusion is that the Deligne conjecture for the corresponding motives is mostly implied by an absolute Hodge version of the Tate conjecture on cycle classes in \(\ell\)-adic cohomology. An unconditional version of this result is the subject of the remaining articles in the series.

The second part of the article describes properties (partly conjectural) of motives associated to self-dual cohomological cuspidal automorphic representations of the adeles of \(\text{GL}(n, \mathbb{Q})\). These motives, realized in the cohomology of Shimura varieties attached to certain unitary groups, are both regular and polarized, hence the theory of the first part applies. The third part of the article uses the Piatetski-Shapiro-Rallis integral representation of the standard \(L\)-function of unitary groups, and techniques developed by Shimura and the author, to compute the critical values of these \(L\)-functions in terms of period integrals. The conclusion is that the Deligne conjecture for the corresponding motives is mostly implied by an absolute Hodge version of the Tate conjecture on cycle classes in \(\ell\)-adic cohomology. An unconditional version of this result is the subject of the remaining articles in the series.

Reviewer: M.Harris (Paris)

##### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

14G35 | Modular and Shimura varieties |