Motivic Cohomology and Arithmetic Intersections

Motivic cohomology provides a universal cohomology theory for algebraic varieties, unifying Chow groups, algebraic $K$-theory, and étale cohomology through a system of realization functors. We present Voevodsky's construction via presheaves with transfers on the Nisnevich site and relate it to Bloch's higher Chow groups $\text{CH}^p(X, n)$. In the second part, we develop arithmetic intersection theory on arithmetic varieties $\mathcal{X} to \text{Spec}(\mathbb{Z})$ following Gillet-Soulé, and discuss how the arithmetic Grothendieck-Riemann-Roch theorem connects heights of cycles to special values of $L$-functions, as predicted by the Beilinson conjectures.