Computations in Stable Motivic Homotopy Theory.

Abstract

This thesis is concerned with the application of certain computational methods from stable algebraic topology in motivic homotopy theory over p-adic fields. My main tools are motivic analogues of the Adams and Adams-Novikov spectral sequences. I determine the coefficients of 2-complete algebraic cobordism and a type of connective algebraic K-theory in the motivic setting. I describe the E_2-term of the motivic Adams-Novikov spectral sequence in terms of the E_2-term of the topological Adams-Novikov spectral sequence and basic arithmetic information. Within this algebra, I discover a motivic analogue of the alpha family and determine its behavior within the motivic Adams-Novikov spectral sequence. This is an ``infinite result" in the stable motivic homotopy groups of the 2-complete sphere spectrum over a p-adic field.