2022/06/13 by Sumit Chandra Mishra, Mishra, Sumit Chandra
Mathematics · #11E72 #12G05 #14G05 #14G27 #14H25 #20G15 #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2206.05911
openalex publication_date 2022/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be a complete discretely valued field with the residue field κ. Assume that cohomological dimension of κ is less than or equal to 1 (for example, κ is an algebraically closed field or a finite field). Let F be the function field of a curve over K. Let n be a squarefree positive integer not divisible by char(κ). Then for any two degree n abelian extensions, we prove that the local-global principle holds for the associated multinorm torus with respect to discrete valuations. Let \mathscrX be a regular proper model of F such that the reduced special fibre X is a union of regular curves with normal crossings. Suppose that κ is algebraically closed with char(κ)≠ 2. If the graph associated to \mathscrX is a tree (e.g. F = K(t)) then we show that the same local-global principle holds for the multinorm torus associated to finitely many abelian extensions where one of the extensions is quadratic and others are of degree not divisible by 4.