Motivic Homotopies

Over global fields, the PI plans to compute with the Adams spectral sequence in the examples of connective algebraic K-theory and motivic Brown-Peterson theory. The approach is reminiscent of the local-to-global philosophy employed in class field theory. A local-to-global principle is a statement asserting that a global field satisfies an arithmetic property if and only if it the same property holds locally in every completion of the global field. The prime examples of such local-to-global principles in clude Hasse's norm theorem in class field theory and the Hasse-Minkowski theorem in quadratic form theory. For the initial example in characteristic zero, the rational numbers, our approach consists of the following steps: (1) Computations for the real nu mbers and the p-adic numbers. (2) Computations for the rational numbers based on the previous two cases This approach can be formalized to global number fields and their completions. In the example of the Gaussian numbers this involves computations with t he complex numbers and degree two extensions of the p-adic numbers. Our program aims at: (3) Complete computations of the algebraic K-groups and cobordism groups of number fields. (4) Partial computations of the motivic stable homotopy groups of number fields. In turn, these steps will allow us to (5) Perform the local-to-global process for the Adams-Novikov spectral sequence of the motivic sphere spectrum over the rational numbers. We envision an extension of the Morel-Voevodsky homotopy theory of sc hemes to the setting of Deligne-Mumford stacks. This is supported by a recent descent theorem for K-theory of Deligne-Mumford stacks. Using this in an equivariant setting, we plan to show the Russell cubic hypersurface in the affine 4-plane is contract ible in the sense of motivic homotopy theory. This suffices to settle a long-standing geometric cancellation conjecture.