Milnor K-group attached to a torus and Birch-Tate conjecture

2008/04/21 by Takao Yamazaki, Yamazaki, Takao
Mathematics · #11R42 #11R70 (Primary) #19F15 (Secondary) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Number Theory (math.NT) #math.KT #math.NT #msc:11R42 #msc:11R70 #msc:19F15

paper · pdf · doi:10.48550/arxiv.0804.3260

16 pages

arxiv created 2008/04/21 · openalex publication_date 2008/04/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We formulate (and prove under a certain assumption) a conjecture relating the order of Somekawa's Milnor K-group attached to a torus T and the value of the Artin L-function attached to the cocharacter group of T (regarded as an Artin representation) at s=-1. The case T=\G_m reduces to the classical Birch-Tate conjecture.

Citations

• Motivic interpretation of Milnor K-groups attached to Jacobian varieties

Related

• A counterexample to generalizations of the Milnor-Bloch-Kato conjecture
• Motivic interpretation of Milnor K-groups attached to Jacobian varieties
• Geometric presentations of Milnor K-groups of certain Artin algebras and Bass-Tate-Kato norms
• Remarks on Milnor K-theory and Tate's conjecture for divisors
• The Milnor K-theory and the Shintani cocycle