Birational classification of fields of invariants for groups of order 128

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• 作者：Akinari Hoshi up>hoshi@math.sc.niigata-u.ac.jp" class="auth_mail" title="E-mail the corresponding authorup>
• 关键词：12F12 ; 13A50 ; 14E08 ; 14F22 ; 16K50 ; 20C10
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：445
• 期：Complete
• 页码：394-432
• 全文大小：690 K

文摘

Let G   be a finite group acting on the rational function field ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si1.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=d40621a648ad7d259c351dc10d3fcf92" title="Click to view the MathML source">C(xub>gub>:g∈G) by ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C-automorphisms ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si3.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=5950616b24412673207b28c42ac9a75f" title="Click to view the MathML source">h(xub>gub>)=xub>hgub> for any ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si35.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=7e6373cf0de51d34947bf30a72dba009" title="Click to view the MathML source">g,h∈G$g,h\in G$. Noether's problem asks whether the invariant field ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si5.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=0afe83a670836462d6b564d8ce794b1c" title="Click to view the MathML source">C(G)=k(xub>gub>:g∈G)up>Gup> is rational (i.e. purely transcendental) over ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C. By Fischer's theorem, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si42.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a549acc255f42c1a2e75557aa2a40332" title="Click to view the MathML source">C(G) is rational over ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G   of order ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si7.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=62eb1f78c9dbca5b4f35f0594269791b" title="Click to view the MathML source">pup>9up> and of order ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si8.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=35cd338edc751d97f19bf6112acb74cd" title="Click to view the MathML source">pup>6up> such that ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si42.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a549acc255f42c1a2e75557aa2a40332" title="Click to view the MathML source">C(G) is not rational over ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C by showing the non-vanishing of the unramified Brauer group: ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si10.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=0208b2cb63885f6490dd5f1c6bf531f8" title="Click to view the MathML source">Brub>nrub>(C(G))≠0, which is an avatar of the birational invariant ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si11.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=360a8383560344747220bd1f8bb8626c" title="Click to view the MathML source">Hup>3up>(X,Z)ub>torsub> given by Artin and Mumford where X   is a smooth projective complex variety whose function field is ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si42.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a549acc255f42c1a2e75557aa2a40332" title="Click to view the MathML source">C(G). For ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si12.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=2ccfd4b2d5dcfa68698041572a3aa942" title="Click to view the MathML source">p=2$p=2$, Chu, Hu, Kang and Prokhorov proved that if G   is a 2-group of order ≤32, then ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si42.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a549acc255f42c1a2e75557aa2a40332" title="Click to view the MathML source">C(G) is rational over ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C. Chu, Hu, Kang and Kunyavskii showed that if G   is of order 64, then ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si42.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a549acc255f42c1a2e75557aa2a40332" title="Click to view the MathML source">C(G) is rational over ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C except for the groups G   belonging to the two isoclinism families ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si13.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=e93bcfe5f78dfde6cc7d1928290115d7" title="Click to view the MathML source">Φub>13ub> with ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si14.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=6360c384d937c824e34a9a1aae12e621" title="Click to view the MathML source">Brub>nrub>(C(G))=0 and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si15.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=7b60f04ef340676575b5258e017bd5e2" title="Click to view the MathML source">Φub>16ub> with ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si16.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=87f61ccea6db6d0e4ed2e0c6c7a548eb" title="Click to view the MathML source">Brub>nrub>(C(G))≃Cub>2ub>. Bogomolov and Böhning's theorem claims that if ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si17.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=ceacb99e996eef59018b91da087f2d45" title="Click to view the MathML source">Gub>1ub> and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si18.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=8391cb52f4de3c33ea4aad8fe3b36429" title="Click to view the MathML source">Gub>2ub> belong to the same isoclinism family, then ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si181.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=466d95c762cee3b9072058d4a339cc31" title="Click to view the MathML source">C(Gub>1ub>) and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si182.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=071c281be16da86b6e15d26105e682de" title="Click to view the MathML source">C(Gub>2ub>) are stably ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C-isomorphic. We investigate the birational classification of ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si42.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a549acc255f42c1a2e75557aa2a40332" title="Click to view the MathML source">C(G) for groups G   of order 128 with ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si10.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=0208b2cb63885f6490dd5f1c6bf531f8" title="Click to view the MathML source">Brub>nrub>(C(G))≠0. Moravec showed that there exist exactly 220 groups G   of order 128 with ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si10.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=0208b2cb63885f6490dd5f1c6bf531f8" title="Click to view the MathML source">Brub>nrub>(C(G))≠0 forming 11 isoclinism families ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si21.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=685c8cb6f410468d2b8af152a884b396" title="Click to view the MathML source">Φub>jub>. We show that if ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si17.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=ceacb99e996eef59018b91da087f2d45" title="Click to view the MathML source">Gub>1ub> and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si18.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=8391cb52f4de3c33ea4aad8fe3b36429" title="Click to view the MathML source">Gub>2ub> belong to ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si15.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=7b60f04ef340676575b5258e017bd5e2" title="Click to view the MathML source">Φub>16ub>, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si22.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=3ca19cc156039c90da66e20fad5e5a0e" title="Click to view the MathML source">Φub>31ub>, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si23.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a105b0c132fdb908aa6dcd7e4e68ce53" title="Click to view the MathML source">Φub>37ub>, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si24.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=cb889e365f1278f912ee7b747db0474b" title="Click to view the MathML source">Φub>39ub>, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si243.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=42d4344161412aa40d9442bfc53fb189" title="Click to view the MathML source">Φub>43ub>, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si244.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=d63062ed4fb7095cc7904d585fc344f4" title="Click to view the MathML source">Φub>58ub>, ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si245.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=ca3466d14ddea0e60495bb5d6b11d10c" title="Click to view the MathML source">Φub>60ub> or ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si246.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=e5d3a5b690fad81daf82120d89d38692" title="Click to view the MathML source">Φub>80ub> (resp. ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si247.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=bfe18903ee463ad6087cc9f1eb65aee9" title="Click to view the MathML source">Φub>106ub> or ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si248.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=38ff84b7ad221ce796234e6ee116e638" title="Click to view the MathML source">Φub>114ub>), then ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si181.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=466d95c762cee3b9072058d4a339cc31" title="Click to view the MathML source">C(Gub>1ub>) and ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si182.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=071c281be16da86b6e15d26105e682de" title="Click to view the MathML source">C(Gub>2ub>) are stably ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si128.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=593e97c57e32b56ae381a048164cadc7" title="Click to view the MathML source">C-isomorphic with ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si31.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a7bed479e44728ca7994650015609ba0" title="Click to view the MathML source">Brub>nrub>(C(Gub>iub>))≃Cub>2ub>. Explicit structures of non-rational fields ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si42.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=a549acc255f42c1a2e75557aa2a40332" title="Click to view the MathML source">C(G) are given for each cases including also the case ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si239.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=cc32b5ac944dadcbdcf2f6b3cdb2aaed" title="Click to view the MathML source">Φub>30ub> with ulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869315003476&_mathId=si33.gif&_user=111111111&_pii=S0021869315003476&_rdoc=1&_issn=00218693&md5=d32fdc72644788c605e73c3ca826ff85" title="Click to view the MathML source">Brub>nrub>(C(G))≃Cub>2ub>×Cub>2ub>.