Abstract :
Abstract: Motives can be seen as universally defined subspaces
of the cohomology of varieties.
The Langlands Program links the theory of motives with
the theory of modular forms. In the Proceedings of the 1991-Seattle
conference on motives, Serre asks the "very hazardeous" question:
Do there exist motives with Galois group isomorphic to
the simple algebraic group G_2 (or E_8)?
Hodge theory implies that such motives of type
G_2 and E_8 cannot occur in the (untwisted) cohomology of Shimura
varieties - which makes them hard to construct.
Using the middle convolution and previous work of Feit, Fong and
Thompson,
we (Stefan Reiter and myself) construct G_2-motives explicitly,
giving a positive answer to Serre's question in the G_2-case.