Rationality problem for fields of invariants : some examples (Researches on isometries from various viewpoints)

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(1)35. 数理解析研究所講究録第2035巻 2017年 35-48. Rationality problem. for fields of invariants:. examples. some. Akinari Hoshi. Department of Mathematics, Niigata University In this. article,. we. give. a. short survey of. some. known results and. our. recent results in. [HKK13], [CHHK15], [Hos16], [HKY16] (with Huah Chu, Shou‐Jen Hu, Ming‐chang Kang, Boris E. Kunyavskii and Aiichi Yamasaki) about rationality problem for fields of invariants and unramified Brauer (cohomology) groups. 1. Introduction. Let k be a field and G be a finite group acting on the rational function field k(x_{g} : g\in G) by k‐automorphisms h(x_{g})=x_{hg} for any g, h\in G We denote the fixed field k(x_{g} : g\in G)^{G} by k(G) Emmy Noether [Noe13, Noe17] asked whether k(G) is rational (= purely transcen‐ dental) over k This is called Noethers problem for G over k and is related to the inverse .. .. .. Galois. ,. to the existence of. problem,. versal G ‐torsors. generic G‐Galois. [MT86], Garibaldi, Merkurjev Colliot‐Thélène and Sansuc [CTS07]).. and Tsfasman. (Fischer [Fis15],. Theorem 1.1. abelian group with exponent. e. k contains. root. a. (see. k ‐rational field extensions. over. primitive e‐th. .. see. (i). k , and to the existence of. [\mathrm{S}\mathrm{a}\mathrm{l}82\mathrm{a}] Swan [Swa83], Manin [GMS03, Section 33.1, page 86. [Swa83,. ,. Theorem. either char k=0. Then. of unity.. over. Saltman. and Serre. also Swan. Assume that. extensions. k(G). or. is k ‐rational.. 6.1]).. finite k=p \paral el e (ii) particular, ￡f(G) is Let G be. char In. ,. a. and. \mathbb{C} ‐rational. Theorem 1.2. field. a. We. (Kuniyoshi [Kun54], [Kun55], [Kun56],. with char now. k=p>0. recall. some. Definition 1.3. Let. and G be. a. finite. see. also Gaschütz. p ‐group. Then. k(G). [Gas59]).. Let k be. is k ‐rational.. relevant definitions of k ‐rationality of fields.. K/k. L/k. finitely generated extensions of fields. (1) (for short, k‐rational) if K is purely transcendental over k i.e. K\simeq k(x_{1}, \ldots, x_{n}) for some algebraically independent elements x_{1} x_{n} over k ; if K is said to be k ‐rational is k for some ‐rational stably algebraically (2) K(y_{1}, \ldots,y_{m}) independent elements y_{1} y_{m} over K ; (3) K and L are said to be stably k ‐isomorphic if K(y_{1}, \ldots, y_{m}) \simeq L(z_{1}, \ldots, z_{n}) for some z_{n} over L ; y_{m} over K and z_{1} algebraically independent elements y_{1} and. K is said to be rational. over. be. k. ,. ,. ,. \cdots. ,. ,. ,. (4) (Saltman, [\mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{b}. \cdots. \cdots. ,. ,. \cdots. ,. a k‐ 3.1]) A in K such that K the field of A there exist a non‐zero is contained quotient algebra (i) (ii) polynomial f\in k[x_{1}, . . . , x_{n}] and k‐algebra homomorphisms $\varphi$:A\rightarrow k[x_{1}, . . . , x_{n}][1/f] and $\psi$:k[x_{1}, \cdots, x_{n}][1/f]\rightarrow A satisfying $\psi$\circ $\varphi$=1_{A;} (5) K is said to be k ‐unirational if k\subset K\subset k(x_{1}, \ldots , x_{n}) for some integer n. ,. Definition. K is said to be retract k ‐rational if there exists ,.

(2) 36. In Saltmans. 3.1]),. original. definition of retract k ‐rationality. ( [\mathrm{S}\mathrm{a}\mathrm{l}82\mathrm{b}. 130], [\mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{b}. page. ,. ,. Def‐. required to be infinite in order to guarantee the existence of sufficiently many k‐specializations. We now assume that k is an infinite field. Then if K and L are stably k ‐isomorphic and K is retract k ‐rational, then L is also retract k‐rational (see [ \mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{b} Proposition 3.6]), and it is not difficult to verify the following implications: inition. base field k is. a. ,. k‐rational. stably. \Rightarrow. retract k‐rational. k‐rational \Rightarrow. k‐unirational.. \Rightarrow. k(G) is retract k‐rational if and only if there exists a generic G‐Galois extension (see [\mathrm{S}\mathrm{a}\mathrm{l}82\mathrm{a} Theorem 5.3], [\mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{b} Theorem 3.12]). In particular, if k is a Hilbertian field, e.g. number field, and k(G) is retract k‐rational, then inverse Galois problem for G over k has a positive answer, i.e. there exists a Galois extension K/k with \mathrm{G}\mathrm{a}1(K/k)\simeq G. Swan [Swa69] gave the first negative solution to Noethers problem. He proved that if p=47 113 or 233, then \mathbb{Q}(C_{p}) is not \mathb {Q}‐rational, where C_{p} is the cyclic group of order prime p by using Masudas idea of Galois descent [Mas55, Mas68]. Note that over. k. ,. ,. ,. ,. Noethers problem for abelian groups was studied extensively by Masuda, Kuniyoshi, Swan, Voskresenskii, Endo and Miyata, etc. Eventually, Lenstra [Len74] gave a necessary. and sufficient condition to Noethers. [Swa83],. survey paper. just. problem. handful of results about Noethers. a. for finite abelian groups. For. [Vos98,. Voskresenskiis book. Section. problem. are. 7]. or. [Hos15].. details,. Swans. see. On the other. obtained when the groups. hand,. are non‐. abelian. Theorem 1.4 group. of degree. (Maeda [Mae89, Theorem, page 418 5. Then k(A_{5}) is k ‐rational.. Let k be. (Serre [GMS03, Chapter IX],. Kang [Kan05]). Let G be a finite group or the generalized quaternion Q_{16} of. Theorem 1.5 with. a. 2‐Sylow subgroup. order 16. Then. \mathbb{Q}(G). which bs. is not. see. also. a. field. and. A5. be the. alternating. order \geq 8. cyclic of stably \mathb {Q} ‐rational.. (Plans [Pla09, Theorem 2 Let A_{n} be the alternating group of degree n If \mathbb{Q}(A_{n}) is rational over \mathbb{Q}(A_{n-1}) In particular, if \mathbb{Q}(A_{n-1}) is \mathb {Q} ‐rational, then so is \mathbb{Q}(A_{n}) Theorem 1.6. n. .. \geq 3 is odd integer, then. .. .. However, From. it is. now. and Theorem with char. on,. 1.2,. k\neq p. Theorem 1.7. \leq p^{4}. and. an. .. open. we we. whether. problem. restrict ourselves to the may focus. on. the. For r‐groups of small. (Chu. of exponent. k(A_{n}). case. is k‐rational for n\geq 6.. case. where G is. where G is. a. a. p\overline{-} group.. By. Theorem 1.1. non‐abelian p‐group and k is. order, the following results. are. a. field. known.. Kang [CKOI]). Let p be any prime and G be a p ‐group of order If k is a field containing a primitive e‐th root of unity, then k(G) is. and e. .. k ‐rational.. Theorem 1.8. (Chu, Hu, Kang and. of exponent e If k\dot{u} .. For. more. recent. a. results,. Saltman introduced. a. see. e.g.. rational.. \Rightarrow retract. a. [CHKP08]).. Let G be. a. e‐th root. of unity,. then. primitive. k‐rational.. group. implications. Hence if. k(G). for. an. of order 32. k(G). [HK10], [Kanll], [KMZ12]. (see Definition 1.3). notion of retract k ‐rationality. ified Brauer group. Recall that the. k‐rational. Prokhorov. field containing. and. is k ‐rational.. and the. unram‐. infinite field k : k-rational \Rightarrow. stably. is not retract k ‐rational, then it is not k‐.

(3) 37. Definition 1.9 of fields. The. (Saltman [ \mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{a}. unramified. 3.1], [Sa185, page 56 \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K/k) of K over k is. Definition. ,. Brauer group. K/k. Let. be. extension. an. defined to be. \displaystyle\mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r}(K/k)=\bigcap_{R}\mathrm{I}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\{ mathrm{B}\mathrm{r}(R)\rightar ow\mathrm{B}\mathrm{r}(K)\} where. \mathrm{B}\mathrm{r}(R)\rightar ow \mathrm{B}\mathrm{r}(K). valuation. is the natural map of Brauer groups and R. R such that k \subset R\subset. rings. ,. finite. (Bogomolov [Bog88,. group and k be. \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (k(G)/k). \dot{u}. In. .. particular, íf k. is. closed. algebraically. an. \dot{u} retract k‐. and K is. field. \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K)=0.. retract k ‐rational, then. a. .. (Saltman [\mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{a}] [Sa185, Proposition 1.8], [SalS7]). If K. 1.10. \mathrm{B}\mathrm{r}(k)\rightar ow^{\sim}\mathrm{B}\mathrm{r}_{\mathrm{n}x}(K). Theorem 1.11. all the discrete. K and K is the quotient field of R We omit k from just \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K) when the base field k is clear from the context.. the notation and write. Proposition rational, then. runs over. Theorem. 3.1],. Saltman. [Sa190,. closed field with char k=0. algebraically isomorphic to the group B_{0}(G) defined by an. Let G be. Theorem 12. or. |G|. char k=p. .. Then. B_{0}(G)=\displaystyle \bigcap_{A}\mathrm{K}\mathrm{e}\mathrm{r}\{\mathrm{r}\mathrm{e}\mathrm{s}:H^{2}(G, \mathb {Q}/\mathb {Z})\rightar ow H^{2}(A, \mathb {Q}/\mathb {Z})\} where A a. cyclic. runs over. group. all the direct. or a. Remark 1.12. For. a. bicyclic subgroups of G (a group product of two cyclic groups).. smooth. projective variety. to the birational invariant. isomorphic [AM72] to provide. is. rational. (see. also. X. over. H^{3}(X, \mathb {Z})_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}. A is called. bicyclic if A. \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K/\mathb {C}). ￠ with function field K,. which. was. used. by Artin and. of k ‐unirational varieties which. elementary examples [Bog88, Theorem 1.1 and Corollary some. is either. Mumford. are. not k-. Following Kunyavskii [KunlO], we call B_{0}(G) the Bogomolov multiplier of G Note that B_{0}(G) is a subgroup of H^{2}(G, \mathbb{Q}/\mathbb{Z}) which is isomorphic to the Schur multiplier H_{2}(G, \mathbb{Z}) of G (see Karpilovsky [Kar87]). Because of Theorem 1.11, we will not distinguish B_{0}(G) and \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (k(G)/k) when k is an algebraically closed field, and char k=0 or char k=p /\}'|G|. Using the Bogomolov multiplier B_{0}(G) Saltman and Bogomolov gave counter‐examples .. ,. to Noethers. problem. Theorem 1.13. algebraically. for non‐aUelian p‐‐groups. over. algebraically. (Saltman [\mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{a}] Bogomolov [Bog88]). ,. closed. field. with char. closed field.. Let p be any. prime and k be. any. k\neq p.. (1) (Saltman [\mathrm{S}\mathrm{a}\mathrm{l}84\mathrm{a} Theorem 3.6]) There exists a meta‐abehan group G of order p^{9} such that B_{0}(G)\neq 0 In particular, k(G) is not (retract, stably) k ‐rational; (2) (Bogomolov [Bog88, Lemma 5.6]) There exists a group G of order p^{6} such that B_{0}(G)\neq 0. In particular, k(G) \dot{u} not (retract, stably) k ‐rational. ,. .. Colliot‐Thélène and group. \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K/k) in [CTO89,. F_{n}^{i,j}(K/k). Definition 1.14 Let. n. Ojanguren [CTO89] generalized the notion of the unramified cohomology H_{\mathrm{n}\mathrm{r} ^{i}(K/k, $\mu$_{n}^{\otimes j}) of degree i \geq 1. to the unramified. be. a. Definition. 1.1].. (Colliot‐Thelène. positive integer. ,. Brauer. that is. and k be. Ojanguren [\mathrm{C}\mathrm{T}\mathrm{O}89| see also [CT95, Sections 2−4]). algebraically closed field with char k=0 or char k=p. and an. ,.

(4) 38. In. K/k. Let. .. cohomology. be. a. function. field,. H_{\mathrm{n}\mathrm{r} ^{i}(K/k, $\mu$_{n}^{\otimes j}). group. that is of K. finitely generated as a field over k The unramified k of degree i\geq 1 is defined to be .. over. H_{\mathrm{n}\mathrm{r} ^{i}(K/k,$\mu$_{n}^{\otimesj})=\displaystyle\bigcap_{R} Image \{H_{\mathrm{e}\mathrm{t} ^{i}(R,$\mu$_{n}^{\otimesj})\rightar owH_{\mathrm{e}\mathrm{t} ^{i}(K,$\mu$_{n}^{\otimesj})\} where R. runs over. K is the. all the discrete valuation. field of R We write. quotient. .. Note that the unramified. such that k\subset R\subset K and. two. are. to the. isomorphic. n\mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K/k)\simeq H_{\mathrm{n}\mathrm{r} ^{2}(K/k, $\mu$_{n}). \dot{u} omorphic, then. .. an algebraically closed field with char k=0 or char k=p\parallel n. Ojanguren [CTO89, Proposition 1.2]) If K and L are stably k‐ H_{\mathrm{n}\mathrm{r} ^{i}(K/k, $\mu$_{n}^{\otimes \mathrm{j} ) \rightar ow^{\sim} H_{\mathrm{n}\mathrm{r} ^{i}(L/k, $\mu$_{n}^{\otimes j}) In particular, K \dot{u} stably k ‐rational,. and. .. H_{\mathrm{n}x}^{i}(K/k, $\mu$_{n}^{\otimes j})=0. ;. (2) ([MerOS, Proposition 2.15], see also [CTO89, Remarque 1.2.2], [CT95, [GS10, Example 5.9]) If K\dot{u} retract k ‐rational, then H_{\mathrm{n}\mathrm{r} ^{i}(K/k, $\mu$_{n}^{\otimes j})=0. Colliot‐Thélène and. field of. a. quadric. Mumford. 0. Section. z. for. a. example. an. result of Suslin. 3] produced. H_{\mathrm{n}\mathrm{r} ^{3}(K, $\mu$_{2}^{\otimes 3})\neq 0. \langle g_{1}g_{2}\rangle. =. gave. \langle\{f_{1}, f_{2}\gg a. example. ,. as a. 2−4],. of not. where K is the function. ,. generalization. \mathbb{C}(x, y, z). of Artin and. H_{\mathrm{n}\mathrm{r} ^{i}(K/k, $\mu$_{p}^{\otimes i})\neq H_{\mathrm{n}\mathrm{r} ^{3}(K/k, $\mu$_{p}^{\otimes 3})\neq 0 and \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K/k). sufficient condition for. of the function field K with. [Sus91]. the first. Sections. the rational function field. over. 2‐fold Pfister form. [AM72]. Peyre [Pey93, Corollary 3]. produced. using. \ll f_{1}, f_{2}. of the type. with three variables x, y, 0 and. Ojanguren [CTO89,. \mathbb{C} ‐rational but \mathbb{C} ‐unirational field K with. stably. n ‐torsion. 1.15. Let k be. (1) (Colliot‐Thélène then. H_{\mathrm{n}\mathrm{r} ^{i}(K, $\mu$_{n}^{\otimes j}). one. when the base field k is clear.. cohomology groups of degree. part of the unramified Brauer group:. Proposition. just. R of rank. rings. =. where K is the function field of. of. product simple algebras of degree p (see [Pey93, Proposition 7 Using a result of Jacob and Rost [JR89], Peyre [Pey93, Proposition 9] also gave an example of H_{\mathrm{n}\mathrm{r} ^{4}(K/k, $\mu$_{2}^{\otimes 4})\neq 0 and \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K/k)=0 where K is the function field of a product of quadrics a. varieties associated to. associated to. a. cyclic. a. some norm. central. 4‐‐fold Pfister form. \ll a_{1}, a_{2}. Take the direct limit with respect to. ,. a4\gg (see. a3,. also. [CT95,. Section. 4.2]).. n:. H^{i}(K/k, \displaystyle \mathbb{Q}/\mathbb{Z}(j) =\lim_{\vec{n} H^{ $\iota$}(K/k, $\mu$_{n}^{\otimes j}) and. we. also define the unramified. cohomology. group. H_{\mathrm{n}x}^{i}(K/k, \displaystyle \mathb {Q}/\mathb {Z}(j) =\bigcap_{R} Image \{H_{\mathrm{e}\mathrm{t} ^{i}(R, \mathbb{Q}/\mathbb{Z}(j) \rightar ow H_{\mathrm{e}'\mathrm{t} ^{i}(K, \mathbb{Q}/\mathbb{Z}(j) \}. Then. we. have. \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (K/k)\simeq H_{\mathrm{n}x}^{2}(K/k, \mathbb{Q}/\mathbb{Z}(1). Peyre [Pey08]. was. able to construct. unramified Brauer group Theorem 1.16. (Peyre [Pey08,. Theorem 3. order. The idea of that. .. example. of. a. field K ,. as. K. =. \mathbb{C}(G). ,. whose. vanishes, butunramified cohomology of degree three does not vanish:. p^{12} such that B_{0}(G)=0 of (retract, stably) \mathbb{C} ‐rational.. p ‐group G. is not. an. Peyres proof is. K_{\max}^{3}/K^{3}\neq 0 (see [Pey08,. to find. a. Let p be any odd and. subgroup. page 210. prime. Then there. H_{\mathrm{n}\mathrm{r} ^{3}(\mathbb{C}(G), \mathbb{Q}/\mathbb{Z})\neq 0 K_{\max}^{3}/K^{3}. of. .. In. exists. a. particular, \mathrm{C}(G). H_{\mathrm{n}\mathrm{r} ^{3}(\mathbb{C}(G), \mathbb{Q}/\mathbb{Z}). and to show.

(5) 39. [Aso13] generalized Peyres argument [Pey93] and established the following theorem (resp. a smooth projective model Y) of the function field of product of quadrics of the type \ll s_{1} s_{n-1}\gg \langle s_{n}) (resp. Rost varieties) over some Asok. for a. a. smooth proper model X. ,. rational function field. .. .. .. =. ,. \mathbb{C} with many variables.. (Asok [Aso13], see also [AMII, Theorem 3] for retract \mathbb{C} ‐rationality). 1]) For any n>0 there exists a smooth projective complex variety. Theorem 1.17. (1) ([Aso13,. over. Theorem. ,. and. so. X is not. (2) ([Aso13,. X. for which H_{\mathrm{n}\mathrm{r} ^{i}(\mathbb{C}(X), $\mu$_{2}^{\otimes i})=0 for each i<n yet H_{\mathrm{n}\mathrm{r} ^{n}(\mathbb{C}(X), $\mu$_{2}^{\otimes n})\neq 0, \mathrm{A}^{1} ‐connected, nor (retract, stably) \mathbb{C} ‐rational;. that is \mathbb{C} ‐unirational,. ,. Theorem. rationally not \mathrm{A}^{1} ‐connected,. connected. 3]). For any. prime l and. complex variety. nor. (retract, stably). n\geq 2. any. Y such that. there exists. ,. H_{\mathrm{n}\mathrm{r} ^{n}(\mathb {C}(Y), $\mu$_{l}^{\otimes n}). \neq 0. .. a. smooth projective. In. Y is. particular,. \mathbb{C} ‐rational.. Namely, the triviality of the unramified Brauer group or the unramified cohomology of higher degree is just a necessary condition of \mathbb{C}‐rationality of fields. It is unknown whether the vanishing of all the unramified cohomologies is a sufficient condition for \mathbb{C} ‐rationality. It is interesting to consider an analog of Theorem 1.17 for quotient varieties V/G e.g. ,. \mathbb{C}(V_{\mathrm{r}\mathrm{e}\mathrm{g} /G)=\mathbb{C}(G). .. Colliot‐Thélène and Voisin. (Colliot‐Thélène. Theorem 1.18 any smooth. [CTV12]. established:. and Voisin. projective complex variety. [CTV12],. X , there \dot{u}. see. [Voi14,. also. Theorem. 6.18]).. For. exact sequence. an. 0\rightar ow H_{\mathrm{n}\mathrm{r} ^{3}(X, \mathbb{Z})\otimes \mathbb{Q}/\mathbb{Z}\rightar ow H_{\mathrm{n}\mathrm{r} ^{3}(X, \mathbb{Q}/\mathbb{Z})\rightar ow \mathrm{T}\mathrm{o}\mathrm{r}\mathrm{s}(Z^{4}(X) \rightar ow 0 where. Z^{4}(X)=\mathrm{H}\mathrm{d}\mathrm{g}^{4}(X, \mathbb{Z})/\mathrm{H}\mathrm{d}\mathrm{g}^{4}(X, \mathbb{Z})_{\mathrm{a}i\mathrm{l}\mathrm{g} and the lower index. algebraic.. are. In. alg means particular, if X. that. we. consider the group. rationally connected,. is. H_{\mathrm{n}\mathrm{r} ^{3}(X, \mathbb{Q}/\mathbb{Z})\simeq Z^{4}(X) Using Peyres. method. of Theorem 1.16 and. gives. of integral Hodge. then. we. classes which. have. .. [Pey08], an. we obtain the following theorem which is an improvement explicit counter‐example to integral Hodge conjecture with the. aid of Theorem 1.18.. (Hoshi, Kang and Yamasaki [HKY16, Theorem 1.4]). Let p be any odd prime. Then there exists a p ‐group G of order p^{9} such that B_{0}(G)=0 and H_{\mathrm{n}\mathrm{r} ^{3}(\mathbb{C}(G), \mathbb{Q}/\mathbb{Z})\neq 0 In particular, \mathrm{C}(G) is not (retract, stably) \mathbb{C} ‐rahonal. Theorem 1.19. .. The. 1.1. case. where G is. a. group of order. p^{5} (p\geq 3). (2), Bogomolov [Bog88, Remark 1] raised a question to classify the p^{6} with B_{0}(G) \neq 0 He also claimed that if G is a p‐‐group of order \leq p^{5}, B_{0}(G)=0 ([Bog88, Lemma 5.6]). However, this claim was disproved by Moravec:. From Theorem 1.13 groups of order. then. Theorem 1.20 0. if. and. .. (Moravec [Mor12,. only if G=G(3^{5}, i). in the GAP database. Section 8. Let G be. with 28\leq i\leq 30 , where. [GAP]. Moreover, if B_{0}(G)\neq 0. ,. a. group. G(3^{5}, i) then. of order 243.. Then. is the i‐th group. B_{0}(G)\simeq C_{3}.. of. B_{0}(G)\neq. order 243.

(6) 40. [Mor12]. Moravec of G and. gave. formula for. a. B_{0}(G) by using. implemented algorithm \mathrm{b}0\mathrm{g}.\mathrm{g}. an. available from his website. www.. in. fmf. uni‐lj.. in. [Mor12,. Table. nonabelian exterior square G\wedge G. \mathrm{s}\mathrm{i}/\sim_{\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c} /\mathrm{b}0\mathrm{g}.\mathrm{g}. groups G of order \leq 729 apart from the orders as. a. computer algebra system GAP. B_{0}(G)\neq 0. 576 and 640 with. 512,. [GAP],. which is. The number of all solvable. .. was. given. 1].. Hoshi, Kang and Kunyavskii [HKK13] determined r‐groups G of order p^{5} with B_{0}(G)\neq 0 for any p\geq 3 It turns out that they belong to the same isoclinism family. .. Definition 1.21. [G, G]. G and. (Hall [Ha140,. For. $\phi$([g, h]) =[g', h']. a. prime. and. subgroup. of G. .. an. for any g, h\in G_{1} with. integer. denote. n , we. .. Z(G). finite group. Let. a. Two p‐groups. G_{1}. and. and. G_{2}. be the center of. $\phi$ [G_{1}, G_{1}]\rightarrow[G_{2}, G_{2}] :. ,. by G_{n}(p). ,. called isoclinic. are. g'\in $\theta$(gZ(G_{1})) h'\in $\theta$(hZ(G_{1})). p^{n} In G_{n}(p) consider an equivalence relation: only if they are isoclinic. Each equivalence class the j‐th isoclinism family is denoted by $\Phi$_{j}.. order and. p and. Let G be. isomorphisms $\theta$:G_{1}/Z(G_{1})\rightarrow G_{2}/Z(G_{2}). if there exist group. that. page 133. be the commutator. such. :. non‐isomorphic groups of G_{1} and G_{2} are equivalent if. the set of all two groups. of. G_{n}(p). is called. an. isoclinism. family,. (resp. p=3 ), there exist 2p+61+\mathrm{g}\mathrm{c}\mathrm{d}\{4,p-1\}+2\mathrm{g}\mathrm{c}\mathrm{d}\{3,p-1\} (resp. 67) $\Phi$_{10} (see [Jam80, groups G of order p^{5} which are classified into ten isoclinism families $\Phi$_{1} Section 4 The main theorem of [HKK13] can be stated as follows: For. p\geq 5. ,. (Hoshi, Kang. .. ... ,. Kunyavskii [HKK13, Theorem 1.12], [Kan14, page 424 Let p be any odd prime and G be a group of order p^{5} Then B_{0}(G) \neq 0 if and only if G belongs to the tsoclinism family $\Phi$_{10} Moreover, if B_{0}(G)\neq 0 then B_{0}(G)\simeq C_{p}. Theorem 1.22. and. .. .. ,. [Kan14, Remark,. 424].. proof of Theorem 1.22 was given by purely algebraic way. There exist exactly 3 groups which belong to $\Phi$_{10} if p=3 i.e. G=G(243, i) with 28\leq i\leq 30 This agrees with Moravecs computational result (Theorem For the last statement,. see. page. The. ,. .. 1.20). For p\geq 5 the exist exactly 1+\mathrm{g}\mathrm{c}\mathrm{d}\{4,p-1\}+\mathrm{g}\mathrm{c}\mathrm{d}\{3,p-1\} groups ([Jam80, page 621 The following result for the k ‐rationality of k(G) supplements Theorem is unknown whether k(G) is k‐rational for groups G which belong to $\Phi$_{7} : ,. which. belong. to. $\Phi$_{10}. (Chu, Hoshi,. Theorem 1.23. G=G(243, i) In. although. it. Kang [CHHK15, Theorem 1.13]). Let G be a group of a field containing a primitive e‐th root of If B_{0}(G) k ‐rational except possibly for the five groups G which belong to $\Phi$_{7}, i.e.. order 243 with exponent. unity, then k(G). 1.20. is. e. Hu and. =0 and k be. .. with 56\leq i\leq 60.. [HKK13]. and. [CHHK15], not only the evaluation of the Bogomolov multiplier B_{0}(G) k(G) but also the k‐isomorphisms between k(G_{1}) and k(G_{2}) for some. and the k ‐rationality of groups. G_{1} and G_{2} belonging. Bogomolov tion. 1.11]. and. to the. Böhning [BB13]. in the affirmative. as. same. gave. follows.. isoclinism an answer. family to the. given. question raised were. as. [HKK13, Ques‐.

(7) 41. (Bogomolov \mathbb{C}(G_{1}) and \mathbb{C}(G_{2}). Theorem 1.24 then. clinic,. H_{\mathrm{n}\mathrm{r} ^{i} (\mathbb{C}(G2), $\mu$_{n}^{\otimes j}) A. Böhning [BB13, Theorem 6 If G_{1} and G_{2} are iso‐ stably \mathb {C} ‐isomorphic. In particular, H_{\mathrm{n}x}^{i}(\mathbb{C}(G_{1}), $\mu$_{n}^{\otimes j}) \rightar ow^{\sim}. .. 1.2] proved. The. 1.2. are. result of Theorem 1.24. partial. Theorem. and. that if. where G is. case. was. G_{1} and G_{2} a. already given by Moravec. Indeed, Moravec [Mor14, isoclinic, then B_{0}(G_{1})\simeq B_{0} (G2).. are. group of order 64. The classification of the groups G of order. Chu, Hu, Kang. and. for groups G with. with. B_{0}(G) \neq. 0 for p= 2. Kunyavskii [CHKK10]. Moreover, they investigated. B_{0}(G)=0. 27 isoclinism families. $\Phi$_{1}. ,. [CHKK10]. main result of. p^{6}. \cdots. .. There exist 267 groups G of order 64 which. $\Phi$_{27} by Hall and Senior. ,. can. was. [HS64] (see. also. obtained. Noethers. classified into. are. [JNO90,. be stated in terms of the isoclinism families. as. by problem. Table I. The. follows.. (Chu, Hu, Kang and Kunyavskii [CHKK10]). Let G=G(2^{6}, i) 1 \leq i\leq 267, of order 64 in the GAP database [GAP]. Theorem (1) ([CHKKIO, 1.8]) B_{0}(G)\neq 0 if and only if G belongs to the bsoclinism family $\Phi$_{16},. Theorem 1.25. ,. be the i‐th group. G=G(2^{6}, i) with 149\leq i\leq 151, 170\leq i\leq 172, 177\leq i\leq 178 or i=182 Moreover, if B_{0}(G)\neq 0 then B_{0}(G)\simeq C_{2} (see [Kan14, Remark, page 424] for this statement); i.e.. .. ,. 0 and k is an quadratically closed field, (2) ([CHKK10, Theorem 1.10]) If B_{0}(G) k(G) is k ‐rational except possibly for five groups which belong to $\Phi$_{13}, i.e. G=G(2^{6}, i) =. then with. 241\leq i\leq 245. For groups G which belong to $\Phi$_{13}, k ‐rationality of k(G) propositions supplement the cases $\Phi$_{13} and $\Phi$_{16} of Theorem. G=G(2^{6},149) page. 2355]. is. given in [HKK14, Proof of Theorem 6.3], see also [CHKKIO, Example 5.11, proof for other cases can be obtained by the similar manner.. Definition 1.26. Let k be. The. act. on. following two proof, the case of. and the. rational function field. (i). is unknown. The 1.25. For the. field. L_{k}^{(0)}. over. a. \neq 2 and k(X_{1},X_{2}, X_{3}, X_{4}, X_{5}, X_{6}) X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}.. field with char k. k with variables. is defined to be. k(X_{1}, X_{2}, X_{3}, X_{4}, X_{5},X_{6})^{H}. where. be the. H=\langle$\sigma$_{1}, $\sigma$_{2}\rangle \simeq C_{2}\times C_{2}. k(X_{1}, X_{2}, X_{3}, X_{4}, X_{5},X_{6}) by k‐automorphisms. X_{2}\displaystyle \mapsto\frac{1}{X_{1}X_{2}X_{3} , X_{3}\mapsto X_{1}, X_{4}\mapsto X_{6}, X_{5}\displaystyle \mapsto\frac{1}{X_{4}X_{5}X_{6} , $\sigma$_{2}:X_{1}\mapsto X_{2}, X_{2}\mapsto X_{1}, X_{3}\displaystyle \mapsto\frac{1}{X_{1}X_{2}X_{3} , X_{6}\displaystyle \mapsto\frac{1}{X_{4}X_{5}X_{6} . $\sigma$_{1}. :. X_{1}\mapsto X_{3},. X_{6}\mapsto X_{4},. X_{4}\mapsto X_{5}, X_{5}\mapsto X_{4},. (ii) by. The field. L_{k}^{(1)} is defined to be k(X_{1}, X_{2}, X_{3}, X_{4})^{\langle $\tau$)} where \langle$\tau$ ). \simeq C_{2}. acts. on. k ‐automorphisms. k(X_{1}, X_{2}, X_{3},X_{4}). $\tau$:X_{1}\displaystyle \mapsto-X_{1}, X_{2}\mapsto\frac{X_{4} {X_{2} ,\cdot X_{3}\mapsto\frac{(X_{4}-1)(X_{4}-X_{1}^{2})}{X_{3} , X_{4}\mapsto X_{4}. Proposition a. group. of. 1.27. ([CHKK10, Proposition 6.3],. order 64 which. belongs. a\mathbb{C} ‐injective homomorphism $\varphi$. particular, \mathrm{C}(G). and. L_{\mathb {C} ^{(0)}. are. :. to. $\Phi$_{13},. L_{\mathb {C} ^{(0)}. \rightarrow. i.e.. see. also. [HY, Proposition 12.5]).. G=G(2^{6}, i). \mathbb{C}(G). such that. stably \mathbb{C} ‐isomorphic. and. with 241\leq i\leq 245. \mathbb{C}(G). is rational. .. over. B_{0}(G)\simeq \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (L_{\mathbb{C} ^{(0)})=0.. Let G be. There exists. $\rho$(L_{\mathb {C} ^{(0)}. .. In.

(8) 42. Proposition Let G be. a. 1.28. group. 170 < i \leq. ([CHKKIO, Example 5.11,. of. order 64 which. 177 \leq. 172,. L_{\mathb {C}^{\overline{(}1) \rightarrow \mathbb{C}(G). i \leq. 178. belongs. or. i. 182. =. page. to .. 2355], [HKK14,. $\Phi$_{16},. There exists. Proof of Theorem. G(2^{6}, i). G=. i.e.. \mathbb{C} ‐injective. a. 151, homomorphism. \mathbb{C}(G) is rational over $\varphi$(L_{\mathb {C} ^{(1)} In particular, \mathrm{C}(G) stably \mathb {C} ‐isomorphic, B_{0}(G) \simeq \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (L_{\mathb {C} ^{(1)} \simeq C_{2} and hence \mathbb{C}(G) and L_{\mathb {C} ^{(1)} are stably) \mathbb{C} ‐rational. $\varphi$. :. Question 1.3. The. such that. .. ([CHKKIO,. 1.29. case. Section. where G is. 6],. see. Tables. database. [GAP]. By using. for \mathrm{i} in [1. we. obtain the. III. I, II,. .. Section 12. Is. L_{k}^{(0)}. L_{\mathb {C} ^{(1)}. not. (retract,. are. k‐rational9. G(2^{7}, i). Moravecs. are. classified into 115 isoclinism families $\Phi$_{1} ,. be the i‐th group of order. algorithm \mathrm{b}0\mathrm{g}.\mathrm{g}. in. [Mor12]. of. 2^{7}. =. GAP,. ... .,. $\Phi$_{115}. 12\mathrm{S} in the GAP e.g.. 2328] do Print ([\mathrm{i}, \mathrm{B}\mathrm{O}\mathrm{G}( SmallGroup ( 128, \mathrm{i}))], |\prime\backslash \mathrm{n}'') ;od;,. following. Theorem 1.30. Let. [HY,. and. group of order 128. a. There exist 2328 groups of order 128 which. ([JNO90,. also. 6.3]).. with 149 \leq i \leq. theorem.. (Moravec [Mor12,. Section 8, Table 1. Let G be. a. group. of order. 128. Then. only if of following B_{0}(G)\neq 0 if (1) G(2^{7}, i) with i=227,228,229,301,324,325,326,541,543,568,570,579,581,626,627,629,667,668, and. G is. one. the. 220 groups:. 670,675,676,678,691,692,693,695,703,704,705,707,724,725,727, 1783, 1784, 1785, 1786, 1864, 1865,. 1866,1867,1880,1881,1882,1893,1894,1903,1904;. (2) G(2^{7}, i) with 1345\leq i\leq 1399 ; (3) G(2^{7}, i) with 242\leq i\leq 247, 265\leq i\leq 269, 287\leq i\leq 293 ; (4) G(2^{7}, i) with 36\leq i\leq 41 ; (5) G(2^{7}, i) with 1924\leq i\leq 1929, 1945\leq i\leq 1951, 1966\leq i\leq 1972, 1983\leq i\leq 1988_{f}. (6) G(2^{7}, i) with 417\leq i\leq 436 ; (7) G(2^{7}, i) with 446\leq i\leq 455 ; (8) G(2^{7}, i) with i=950 951, 952, 975, 976, 977, 982, 983, 987; (9) G(2^{7}, i) with i=144 145; (10) G(2^{7}, i) with i=138 139; (11) G(2^{7}, i) with 1544\leq i\leq 1577. Moreover, if G is a group in (1)‐(10) (resp. (11)), then B_{0}(G)\simeq C_{2} (resp. C_{2}\times C_{2} ). ,. ,. ,. By [JNO90,. Tables. I, II, III],. we can. get the classification of 115 isoclinism families for. groups G of order 128 in terms of the GAP database. [GAP],. see. [Hos16,. Table. 2]. Using this,. correspond (1)-(11) $\Phi$_{16}, $\Phi$_{31}, $\Phi$_{37}, $\Phi$_{39}, $\Phi$_{43}, $\Phi$_{58}, $\Phi$_{60}, $\Phi$_{80}, $\Phi$_{106}, $\Phi$_{114}, $\Phi$_{30} respectively:. we see. that the groups. as. in. of Theorem 1.30. to the isoclinism families. Let G be a group of order 128. Then Corollary 1.31 (Moravec [Mor12, Section 8, Table 1 and G to the bsoclinism belongs family $\Phi$_{16}, $\Phi$_{30}, $\Phi$_{31}, $\Phi$_{37}, $\Phi$_{39}, $\Phi$_{43}, only if B_{0}(G) \neq 0 if $\Phi$_{58}, $\Phi$_{60}, $\Phi$_{80}, $\Phi$_{106} or $\Phi$_{114} Moreover, if B_{0}(G)\neq 0 then .. ,. B_{0}(G)\simeq\left\{ begin{ar y}{l C_{2}&\mathrm{i}\mathrm{f}G\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{o}$\Phi$_{16},$\Phi$_{31},$\Phi$_{37},$\Phi$_{39},$\Phi$_{43},$\Phi$_{\mathrm{S}8,$\Phi$_{60},$\Phi$_{80},$\Phi$_{106}\mathrm{o}\mathrm{}$\Phi$_{1 4},\ C_{2}\timesC_{2}&\mathrm{i}\mathrm{f}G\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{o}$\Phi$_{30}. \end{ar y}\right. In. particular, \mathbb{C}(G). \dot{u} not. (retract, stably). \mathbb{C} ‐rational..

(9) 43. Table 1: Isoclinism families. $\Phi$_{j}. (stably). It is natural to ask the. for groups G of order 128 with. birational classification of. particular, what happens to \mathbb{C}(G) with B_{0}(G)\neq 0 ? 1.33) gives a partial answer to this question. 128. In. Deflnition 1.32. Let k be. functiÒn. rational. field. over. a. field with char. k with variables. k\neq 2. The. \mathrm{C}(G). B_{0}(G)\neq 0. for groups G of order. following. theorem. (Theorem. and k (X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, X7) be the. X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, X_{7}.. (i) The field L_{k}^{(2)} is defined to be k(X_{1}, X_{2}, X_{3}, X_{4}, X5, X_{6})^{( $\rho$\rangle} k(X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}) by k‐automorphisms. where. \langle $\rho$\rangle\simeq C_{4}. acts. on. $\rho$:X_{1}\mapsto X_{2}, X_{2}\mapsto-X_{1} , X_{3}\mapsto X_{4}, X_{4}\mapsto X_{3},. X_{5}\displaystyle \mapsto X_{6}, X_{6}\mapsto\frac{(X_{1}^{2}X_{2}^{2}-1)(X_{1}^{2}X_{3}^{2}+X_{2}^{2}-X_{3}^{2}-1)}{X_{5} . (ii). The field. acts. L_{k}^{(3)} is defined to be k(X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, X_{7})^{\langle$\lambda$_{1},$\lambda$_{2})} where \langle$\lambda$_{1}, $\lambda$_{2}\rangle\simeq C_{2}\times C_{2}. k (X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, X7). on. by k‐automorphisms. $\lambda$_{1}:X_{1}\displaystyle \mapsto X_{1}, X_{2}\mapsto\frac{X_{1} {X_{2} , X_{3}\mapsto\frac{1}{X_{1}X_{3} , X_{4}\mapsto\frac{X_{2}X_{4} {X_{1}X_{3} ,. X_{5}\displaystyle \mapsto-\frac{X_{1}X_{6}^{2}-1}{X_{5} , X_{6}\mapsto-X_{6}, X_{7}\mapsto X_{7}, $\lambda$_{2}:X_{1}\displaystyle \mapsto\frac{1}{X_{1} , X_{2}\mapsto X_{3}, X_{3}\mapsto X_{2}, X_{4}\mapsto\frac{(X_{1}X_{6}^{2}-1)(X_{1}X_{7}^{2}-1)}{X_{4} , X_{5}\mapsto-X_{5}, X_{6}\mapsto-X_{1}X_{6}, X_{7}\mapsto-X_{1}X_{7}. Theorem 1.33. B_{0}(G)\neq 0. .. (Hoshi [Hos16, Theorem 1.31]). Let G be a group of order \mathrm{C}(G) and L_{\mathb {C} ^{(m)} are stably \mathbb{C} ‐isomorphic where. m=. In. particular,. L_{\mathb {C} ^{(1)}, L_{\mathb {C} ^{(2)} For. =. \{. 1. if G. belongs. to. $\Phi$_{16}, $\Phi$_{31}, $\Phi$_{37}, $\Phi$_{39}, $\Phi$_{43}, $\Phi$_{58}, $\Phi$_{60}. 2. if G. belongs. to. $\Phi$_{106}. 3. if G. belongs. to. $\Phi$_{30}.. or. or. $\Phi$_{80},. $\Phi$_{114},. \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r}(L_{\mathb {C}^{(1)} \simeq\mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r} (L_{\mathb {C} ^{(2)} \simeq C_{2} and \mathrm{B}\mathrm{r}_{\mathrm{n}\mathrm{r}(L_{\mathb {C}^{(3)} \simeq C_{2} \times C_{2} and hence the fields not (retract, stably) \mathbb{C} ‐rational. L_{\mathb {C} ^{(3)} 1 2, the fields L_{\mathb {C} ^{(m)} and L_{\mathb {C} ^{(3)} not stably \mathb {C} ‐isomorphic because their. and. m. 128. Assume that. Then. are. are. ,. ramified Brauer groups. un‐. do not know whether the fields. ramified. isomorphic. However, stably \mathbb{C} ‐isomorphic. If not, it is interesting to evaluate the higher un‐ cohomologies. Unfortunately, a useful formula like Bogomolovs formula (Theorem. 1.11). Moravecs formula. L_{\mathb {C} ^{(1)}. and. or. L_{\mathb {C} ^{(2)}. cohomologies.. are. not. we. are. [Mor12,. Section. 3]. for. B_{0}(G). is unknown for. higher. unramified.

(10) 44. Theorem 1.33 gives another proof of B_{0}(G) \simeq C_{2} to Theorem 1.30 when G belongs to $\Phi$_{16}, $\Phi$_{31}, $\Phi$_{37}, $\Phi$_{39}, $\Phi$_{43}, $\Phi$_{58}, $\Phi$_{60} or $\Phi$_{80} Especially, this proof is based on the result of order .. 64 for $\Phi$_{16}. (Theorem 1.25). Although. Theorem 1.33. further work towards of. \mathrm{C}(G). and it does not. gives only. a more. depend. on. the computer calculations of GAP.. the first step, the author. complete understanding. hopes. (stably). of the. that it will stimulate. birational classification. for non‐aUelian groups G.. References. [AM72]. M. are. [Aso13]. A.. Artin, D. Mumford, Some elementary examples of unirational not rational, Proc. London Math. Soc. (3) 25 (1972) 75‐95. Asok, Rationality problems. pos. Math. 149. (2013). A.. [Bog88]. F. A.. .. Bogomolov,. (Russian). The Brauer group. F. A.. Bogomolov,. C.. (1988). Böhning,. H. J.. [CHKKIO]. and. algebraic. h‐. spaces. 455‐485.. Isoclinism and stable. ucts, Birational geometry, rational. York,. Com‐. of linear representations, (1987) 485‐516, 688. English trans‐. of quotient. Izv. Akad. Nauk SSSR Ser. Mat. 51. lation: Math. USSR‐Izv. 30. [CHHK15]. conjectures of Milnor and Bloch‐Kato,. 1312‐1326.. Asok, $\Gamma$ Morel, Smooth varieties up to \mathrm{A}^{1} ‐homotopy cobordisms, Adv. Math. 227 (2011) 1990‐2058.. [AMII]. [BB13]. and. varieties which. curves, and. cohomology of wreath prod‐. arithmetic, 57‐76, Springer,. New. 2013.. Chu,. S.‐J.. Algebra. Chu, ramified. H.. Hu,. 442. S.‐J.. A.. Hoshi,. (2015). Hu,. M.. Kang,. Noethers. problem for. groups. of. order. 243,. 233‐259.. Kang,. M.. Brauer groups. for. B. E.. Kunyavskii, Noethers problem and the un‐ of order 64, Int. Math. Res. Not. IMRN 2010. groups. 2329‐2366.. [CHKP08.] [CKOI]. [CT95]. H.. Chu,. S.‐J.. Hu,. M.. Kang,. Y. G.. (2008). Prokhorov,. Noethers. order 32, J.. Algebra. H.. Kang, Rationality ofp‐group actions,. Chu,. J.‐L.. M.. 320. Colliot‐Thélène,. problem for. groups. of. 3022‐3035.. Birational. J.. Algebra. 237. (2001). 673‐690.. invariants, purity and the Gersten conjecture,. \mathrm{K} ‐theory and. algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 1‐64, Proc. Sympos. Pure Math., 58, Part 1, Amer. Math. Soc., Providence, RI, 1995.. [CTO89]. J.‐L.. au‐delà de. [CTS07]. J.‐L.. Ojanguren, Variétés unirationnelles non rationnelles: lexample dArtin et Mumford, Invent. Math. 97 (1989) 141‐158.. Colliot‐Thélène,. Colliot‐Thélène,. M.. J.‐J.. Sansuc,. The. rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Algebraic groups and homogeneous spaces, 113‐186, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007..

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