2018/02/01 by Vladimir Chernousov, Chernousov, Vladimir I., Andrei S. Rapinchuk +3
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1802.00299
openalex publication_date 2018/02/01 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We prove that the genus of a finite-dimensional division algebra is finite\nwhenever the center is a finitely generated field of any characteristic. We\nalso discuss potential applications of our method to other problems, including\nthe finiteness of the genus of simple algebraic groups of type textsfG2.\nThese applications involve the double cosets of adele groups of algebraic\ngroups over arbitrary finitely generated fields: while over number fields these\ndouble cosets are associated with the class numbers of algebraic groups and\nhence have been actively analyzed, similar question over more general fields\nseem to come up for the first time. In the Appendix, we link the double cosets\nwith check rm Cech cohomology and indicate connections between certain\nfiniteness properties involving double cosets (Condition (T)) and Bass's\nfiniteness conjecture in K-theory.\n