Periods of Hecke characters. (English) Zbl 0659.14001

Schappacher, Norbert

Lecture Notes in Mathematics, 1301. Berlin etc.: Springer-Verlag. XV, 160 p.; DM 28.50 (1988).

The present article is devoted to the motivic theory for several objects around the periods of Hecke characters.
The objects are Shimura-Taniyama’s theorem for complex multiplication (I,§ 1) absolute Hodge cycles (I,§ 2), Hecke characters (I,§§ 3-5), Taniyama group (I,§ 6), Jacobi sum (I,§ 7), Chowla-Selberg formula (III) and modular forms (V).
The Chapter II is the technical heart of this article. Chapter IV mainly concerns with Shimura’s relations between periods of abelian varieties of CM type and generalizations of the Chowla-Selberg formula to abelian fields from the view point of motifs.

Reviewer: K.Katayama

MSC:

14A20	Generalizations (algebraic spaces, stacks)
14G25	Global ground fields in algebraic geometry
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14K22	Complex multiplication and abelian varieties

Keywords:

motivic theory; periods of Hecke characters; complex multiplication; Taniyama group; Jacobi sum; Chowla-Selberg formula