Birational classification for algebraic tori

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 $[\hat{T}]^{fl}$ as a $\mathbb{Z}_p[{\rm Syl}_p(G)]$-lattice and prove that they are faithful and indecomposable $\mathbb{Z}_p[{\rm Syl}_p(G)]$-lattices unless it vanishes for $p=2$ (resp. $p=2,3$) in dimension $3$ (resp. $4$) via $p$-adic analysis. The $\mathbb{Z}_p$-ranks of them are also given. Third, we give a necessary and sufficient condition for which two not stably rational algebraic tori $T$ and $T^\prime$ of dimensions $3$ (resp. $4$) are stably birationally equivalent in terms of the splitting fields and the weak stably equivalent classes of $T$ and $T^\prime$. In particular, the splitting fields of them should coincide if $\hat{T}$ and $\hat{T}^\prime$ are indecomposable. Forth, for each $7$ cases of not stably but retract rational algebraic tori of dimension $4$, we find an algebraic torus $T^\prime$ of dimension $4$ which satisfies that $T\times_k T^\prime$ is stably rational. Finally, we give a criteria to determine whether two algebraic tori $T$ and $T^\prime$ of general dimensions are stably birationally equivalent when $T$ (resp. $T^\prime$) is stably birationally equivalent to some algebraic torus $T^{\prime\prime}$ of dimension up to $4$.