Distributions and Euler systems for the general linear group

TL;DR

This paper rigorously establishes that tensor products of distributions form Euler systems for GL_d, generalizing known cases and applying this to construct motivic cohomology elements for Drinfeld modular schemes.

Contribution

It generalizes the distribution property and norm relations of Euler systems from GL_1 and GL_2 to GL_d, and applies this to arithmetic of Drinfeld modular schemes.

Findings

Proves tensor products of distributions form Euler systems for GL_d.

Constructs motivic cohomology elements satisfying norm relations.

Provides a group theoretic formulation of Tamagawa's conjecture.

Abstract

The main aim is to give a rigorous statement and proof of the slogan "the d-fold tensor product of distributions is an Euler system for GL_d". Of the few known examples of Euler systems, we look at those of cyclotomic units and of Beilinson-Kato elements. The cyclotomic units satisfy distribution property, and this is the key to the proof of the norm relation property for GL_1. For the Beilinson-Kato elements, the Siegel units satisfy distribution property, and the 2-fold tensor product, giving rise to elements in the K_2 of modular curves, satisfies the norm relation for GL_2. We make this common property clear, generalizing everything to GL_d. As an application (our main arithmetic result), we construct elements in the motivic cohomology of Drinfeld modular schemes with integral coefficient and show that the norm relation common to Euler systems (i.e., the norm of one element is described using the local L-factor times another element) hold. We use the language of Y-sites which was introduced in [Kon-Ya3] in order to simplify the computation common to the theory of automorphic forms. Instead of double cosets, we work more systematically with torsion modules and Q-morphisms between them. The idea is that torsion modules are level structures, and Q-morphisms induce morphisms between some moduli spaces. Chapters 1 and 3 serve as a sequel: further generalities on Y-sites, more examples of Y-sites, and proofs of some statements in loc.cit are given. An application is also given: we provide a group theoretic formulation of a conjecture of Tamagawa on affine curves over an algebraic closure of a finite field.