Weil cohomology theories and their motivic Hopf algebroids

In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing a previous construction of the author for the Betti cohomology. The main result proven in the paper is the connectivity of the motivic Hopf algebroids associated to the classical Weil cohomology theories.