2021/12/04 by Akinari Hoshi, Hoshi, Akinari, Aiichi Yamasaki +1
Mathematics · #11E72 #12F20 #13A50 #14E08 #20C10 #20G15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2112.02280
openalex publication_date 2021/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
We give a stably birational classification for algebraic tori of dimensions 3 and 4 over a field k. First, we define the weak stably equivalence of algebraic tori and show that there exist 13 (resp. 128) weak stably equivalent classes of algebraic tori T of dimension 3 (resp. 4) which are not stably rational by computing some cohomological stably birational invariants, e.g. the Brauer-Grothendieck group of X where X is a smooth compactification of T, provided by Kunyavskii, Skorobogatov and Tsfasman. We make a procedure to compute such stably birational invariants effectively and the computations are done by using the computer algebra system GAP. Second, we define the p-part of the flabby class [T]fl as a ℤp[\rm Sylp(G)]-lattice and prove that they are faithful and indecomposable ℤp[\rm Sylp(G)]-lattices unless it vanishes for p=2 (resp. p=2,3) in dimension 3 (resp. 4) via p-adic analysis. The ℤp-ranks of them are also given. Third, we give a necessary and sufficient condition for which two not stably rational algebraic tori T and T^′ of dimensions 3 (resp. 4) are stably birationally equivalent in terms of the splitting fields and the weak stably equivalent classes of T and T^′. In particular, the splitting fields of them should coincide if T and T^′ are indecomposable. Forth, for each 7 cases of not stably but retract rational algebraic tori of dimension 4, we find an algebraic torus T^′ of dimension 4 which satisfies that T×k T^′ is stably rational. Finally, we give a criteria to determine whether two algebraic tori T and T^′ of general dimensions are stably birationally equivalent when T (resp. T^′) is stably birationally equivalent to some algebraic torus T′′ of dimension up to 4.