Galois modules and Shimura varieties. (Galoismoduln und Shimura-Varietäten.) (German) Zbl 1002.11053

Harder, G.

Jahresber. Dtsch. Math.-Ver. 101, No. 1, 26-46 (1999).

This is the extended version of the author’s talk at the DMV meeting in Jena 1996. The author’s aim is to give a broad audience an idea of some of the new developments in number theory, especially the use of Shimura varieties in the construction of motivic Galois modules with prescribed ramification.
First the author explains basic notions from algebraic number theory, then he treats ‘motivic’ Galois modules, mixed Tate motives, Shimura varieties, the Taniyama conjecture and Wiles’ famous theorem. In the final section he reviews the construction of mixed Tate motives in the Eisenstein cohomology of modular curves.
This excellent review article (in German) offers a large number of hints to related and ongoing research giving thereby a most valuable survey not only for the non-specialist mathematicians.

Reviewer: O.Ninnemann (Berlin)

MSC:

11G18	Arithmetic aspects of modular and Shimura varieties
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G09	Drinfel’d modules; higher-dimensional motives, etc.
11G40	\(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G35	Modular and Shimura varieties

Keywords:

\(L\)-functions of elliptic curves; modularity of elliptic curves; Shimura varieties; motivic Galois modules; mixed Tate motives; Taniyama conjecture; Wiles’ famous theorem