Arakelov motivic cohomology I

TL;DR

This paper develops a new Arakelov-theoretic motivic cohomology for schemes over arithmetic rings, aiming to connect with conjectures on special values of L-functions, and establishes foundational properties and theorems in this framework.

Contribution

It introduces a novel cohomology theory integrating Arakelov geometry with motivic homotopy theory, including formal properties and a higher arithmetic Riemann-Roch theorem.

Findings

Established pullback and pushforward formalism

Proved localization sequences and h-descent

Derived a higher arithmetic Riemann-Roch theorem

Abstract

This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $h$-descent. We round off the picture with a purity result and a higher arithmetic Riemann-Roch theorem.