Motivic Periods and Motivic Sheaves on Stacks

Part I: We give an explicit construction of the p-adic de Rham comparison isomorphism for 1-motives. In particular, we prove that our construction recovers Beilinson's comparison isomorphism and is functorial with respect to morphisms of rigid 1-motives. In the second part, we construct a new ring of motivic p-adic periods using the formalism of rigid analytic motives. Furthermore, we explain the relation between these motivic periods and the periods of the classical p-adic de Rham comparison isomorphism. Finally, we apply our construction to explicitly compute the period pairing for several examples of 1-motives and curves. Part II: We give a construction of the (un-)stable motivic homotopy category of an algebraic stack in the spirit of Morel-Voevodsky. We prove that this new construction agrees with the stable motivic homotopy category defined by Chowdhury. As an application, we prove a version of Bachmann's spectral rigidity theorem for algebraic stacks. Moreover, we extend the construction of the framed motivic homotopy category to algebraic stacks and prove the Reconstruction Theorem in this setting. Finally, we discuss an extension of the formalism of cocomplete coefficient systems of Drew-Gallauer to the setting of algebraic stacks.