Birational classification for algebraic tori

Abstract

We give a stably birational classification for algebraic 𝓀-tori of dimensions 3 and 4 over a field 𝓀. Kunyavskii [Kun90] proved that there exist 15 not stably 𝓀-rational cases among 73 cases of algebraic 𝓀-tori of dimension 3. Hoshi and Yamasaki [HY17] showed that there exist exactly 487 (resp. 7, resp. 216) stably 𝓀-rational (resp. not stably but retract 𝓀-rational, resp. not retract 𝓀-rational) cases of algebraic 𝓀-tori of dimension 4. First, we define the weak stably 𝓀-equivalence of algebraic 𝓀-tori and show that there exist 13 (resp. 128) weak stably 𝓀-equivalent classes of algebraic 𝓀-tori T of dimension 3 (resp. 4) which are not stably 𝓀-rational by computing some cohomological stably birational invariants, e.g. the Brauer-Grothendieck group of 𝘟 where 𝘟 is a smooth 𝓀-compactification of 𝘛, provided by Kunyavskii, Skorobogatov and Tsfasman [KST89]. 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[𝒮𝓎ₚ(𝘎)]-lattice and prove that they are faithful and indecomposable ℤₚ[𝒮𝓎ₚ(𝘎)]-lattices unless it vanishes for 𝘱 = 2 (resp. 𝘱 = 2, 3) in dimension 3 (resp. 4). The ℤₚ-ranks of them are also given. Third, we give a necessary and sufficient condition for which two not stably 𝓀-rational algebraic k-tori 𝘛 and 𝘛′ of dimensions 3 (resp. 4) are stably birationally 𝓀-equivalent in terms of the splitting fields and the weak stably 𝓀-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 𝓀-rational algebraic 𝓀-tori of dimension 4, we find an algebraic 𝓀-torus 𝘛′ of dimension 4 which satisfies that 𝘛×[𝓀]𝘛′ is stably 𝓀-rational. Finally, we give a criteria to determine whether two algebraic 𝓀-tori 𝘛 and 𝘛′ of general dimensions are stably birationally 𝓀-equivalent when 𝘛 (resp. 𝘛′) is stably birationally 𝓀-equivalent to some algebraic 𝓀-torus T″ of dimension up to 4. This is a joint work with Aiichi Yamasaki (arXiv: 2112.02280).