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日時：2022年4月22日(金) 15:00〜16:30

場所：新潟大学理学部A棟523室（大セミナー室）+ Zoom

講演者: 星 明考（新潟大学）

タイトル：Birational classification for algebraic tori (I)

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アブストラクト：
https://arxiv.org/abs/2112.02280
の内容を解説します．以下，arXivのアブストラクトです：

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 [\widehat{T}]^{fl} as
a Z_p[Sy_p(G)]-lattice and prove that they are faithful and indecomposable
Z_p[Sy_p(G)]-lattices unless it vanishes for p=2 (resp. p=2,3) in
dimension 3 (resp. 4). The Z_p-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 \widehat{T} and \widehat{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\times_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
of dimension up to 4.

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世話人：小島秀雄、高橋剛、星明考