\(L\)-functions and periods of adjoint motives. (English) Zbl 1319.11028

Harris, Michael

Algebra Number Theory 7, No. 1, 117-155 (2013).

Summary: The article studies the compatibility of the refined Gross-Prasad (or Ichino-Ikeda) conjecture for unitary groups, due to Neal Harris, with Deligne’s conjecture on critical values of \(L\)-functions. When the automorphic representations are of motivic type, it is shown that the \(L\)-values that arise in the formula are critical in Deligne’s sense, and their Deligne periods can be written explicitly as products of Petersson norms of arithmetically normalized coherent cohomology classes. In some cases this can be used to verify Deligne’s conjecture for critical values of adjoint type (Asai) \(L\)-functions.

MSC:

11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70	Representation-theoretic methods; automorphic representations over local and global fields
14G35	Modular and Shimura varieties
11G09	Drinfel’d modules; higher-dimensional motives, etc.

Keywords:

adjoint \(L\)-functions; automorphic forms; motives; Ichino-Ikeda conjecture; periods