The Gauss and Jacobi sums are finite field analogue of the gamma and beta functions. They admit geometric interpretation in terms of the Artin-Schreier curves and Fermat varieties. We refine this interpretation to the level of motives: we construct Chow motives that are related to the gamma/beta values (resp. Gauss/Jacobi sums) via the Hodge (resp. etale) realization. We then exhibit relations between these motives, which can be regarded as (and yield a new proof of) the classical formulas due to Davenport-Hasse. These results are applied to investigate the group of invertible motives over a finite field. (No knowledge of motives is assumed. This is joint work with Noriyuki Otsubo.)
