| Résume||A classical construction in complex algebraic geometry, due to Kuga and Satake, associates to any K3 surface an abelian variety, with a precise relationship between their Hodge structures. Variations on the original construction have arisen over the years, but always depending on the very special nature of the Hodge theory of abelian varieties. After reviewing some arithmetic evidence that the Kuga-Satake construction should be simply the first case of a general motivic phenomenon, I will give some examples, having nothing to do with abelian varieties, arising from the study of rigid local systems on the punctured line. One set of examples will rely on Katz's theory of middle convolution.
The other will rely on interpreting an elementary construction for automorphic representations in terms of geometric Langlands theory.|