Essential dimension, spinor groups, and quadratic forms

Abstract

We prove that the essential dimension of the spinor group $\mathbf{Spin}_n$ grows exponentially with $n$ and use this result to show that quadratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected.

  • [adams] J. F. Adams, Lectures on Exceptional Lie Groups, Chicago, IL: University of Chicago Press, 1996.
    @book {adams, MRKEY = {1428422},
      AUTHOR = {Adams, J. F.},
      TITLE = {Lectures on Exceptional {L}ie Groups},
      SERIES = {Chicago Lectures in Mathematics},
      PUBLISHER = {University of Chicago Press},
      ADDRESS = {Chicago, IL},
      YEAR = {1996},
      PAGES = {xiv+122},
      ISBN = {0-226-00526-7; 0-226-00527-5},
      MRCLASS = {22-01 (22E10)},
      MRNUMBER = {98b:22001},
      MRREVIEWER = {William M. McGovern},
      ZBLNUMBER = {0866.22008},
      }
  • [ap] E. M. Andreev and V. L. Popov, "The stationary subgroups of points in general position in a representation space of a semisimple Lie group," Funkcional. Anal. i Priložen., vol. 5, iss. 4, pp. 1-8, 1971.
    @article {ap, MRKEY = {0291174},
      AUTHOR = {Andreev, E. M. and Popov, V. L.},
      TITLE = {The stationary subgroups of points in general position in a representation space of a semisimple {L}ie group},
      JOURNAL = {Funkcional. Anal. i Priložen.},
      FJOURNAL = {Akademija Nauk SSSR. Funkcional\cprime nyi Analiz i ego Priloženija},
      NOTE = {In Russian; translated in {\it Functional Anal. Appl} {\bf 5} (1971), 265--271},
      VOLUME = {5},
      YEAR = {1971},
      NUMBER = {4},
      PAGES = {1--8},
      ISSN = {0374-1990},
      MRCLASS = {20G05 (22E50)},
      MRNUMBER = {45 \#268},
      MRREVIEWER = {E. Vinberg},
      ZBLNUMBER = {0246.22017},
      }
  • [bf1] G. Berhuy and G. Favi, "Essential dimension: a functorial point of view (after A. Merkurjev)," Doc. Math., vol. 8, pp. 279-330, 2003.
    @article {bf1, MRKEY = {2029168},
      AUTHOR = {Berhuy, Gr{é}gory and Favi, Giordano},
      TITLE = {Essential dimension: a functorial point of view (after {A}. {M}erkurjev)},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      VOLUME = {8},
      YEAR = {2003},
      PAGES = {279--330},
      ISSN = {1431-0635},
      MRCLASS = {11E72 (12G05 14L30)},
      MRNUMBER = {2004m:11056},
      MRREVIEWER = {Ryan Skip Garibaldi},
      ZBLNUMBER={1101.14324},
      }
  • [bur] Go to document J. Buhler and Z. Reichstein, "On the essential dimension of a finite group," Compositio Math., vol. 106, iss. 2, pp. 159-179, 1997.
    @article {bur, MRKEY = {1457337},
      AUTHOR = {Buhler, J. and Reichstein, Z.},
      TITLE = {On the essential dimension of a finite group},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {106},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {159--179},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {12F10 (12E05 14E05 14L30)},
      MRNUMBER = {98e:12004},
      MRREVIEWER = {Helmut V{ö}lklein},
      DOI = {10.1023/A:1000144403695},
      ZBLNUMBER = {0905.12003},
      }
  • [BrosnanAB] Go to document P. Brosnan, "The essential dimension of a $g$-dimensional complex abelian variety is $2g$," Transform. Groups, vol. 12, iss. 3, pp. 437-441, 2007.
    @article {BrosnanAB, MRKEY = {2356317},
      AUTHOR = {Brosnan, Patrick},
      TITLE = {The essential dimension of a {$g$}-dimensional complex abelian variety is {$2g$}},
      JOURNAL = {Transform. Groups},
      FJOURNAL = {Transformation Groups},
      VOLUME = {12},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {437--441},
      ISSN = {1083-4362},
      MRCLASS = {14K05 (14L99)},
      MRNUMBER = {2008g:14073},
      MRREVIEWER = {Nicolae Manolache},
      DOI = {10.1007/s00031-006-0045-0},
      ZBLNUMBER = {1127.14043},
      }
  • [brv] P. Brosnan, Z. Reichstein, and A. Vistoli, Essential dimension and algebraic stacks, 2007.
    @misc{brv,
      author= {Brosnan, Patrick and Reichstein, Z. and Vistoli, A.},
      TITLE={Essential dimension and algebraic stacks},
      YEAR={2007},
      ARXIV={math/0701903v1},
      }
  • [cs] Go to document V. Chernousov and J. Serre, "Lower bounds for essential dimensions via orthogonal representations," J. Algebra, vol. 305, iss. 2, pp. 1055-1070, 2006.
    @article {cs, MRKEY = {2266867},
      AUTHOR = {Chernousov, Vladimir and Serre, Jean-Pierre},
      TITLE = {Lower bounds for essential dimensions via orthogonal representations},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {305},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {1055--1070},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20G15 (14L30)},
      MRNUMBER = {2007i:20070},
      MRREVIEWER = {Zinovy Reichstein},
      DOI = {10.1016/j.jalgebra.2005.10.032},
      ZBLNUMBER = {05078318},
      }
  • [garibaldi] S. Garibaldi, "Cohomological invariants: exceptional groups and spin groups," Mem. Amer. Math. Soc., vol. 200, 2009.
    @article{garibaldi,
      author = {Garibaldi, Skip},
      TITLE={Cohomological invariants: exceptional groups and spin groups},
      JOURNAL={Mem. Amer. Math. Soc.},
      VOLUME={200},
      YEAR={2009},
      MRNUMBER={2528487},
      }
  • [gms] S. Garibaldi, A. Merkurjev, and J. Serre, Cohomological Invariants in Galois Cohomology, Providence, RI: Amer. Math. Soc., 2003.
    @book {gms, MRKEY = {1999383},
      AUTHOR = {Garibaldi, Skip and Merkurjev, Alexander and Serre, Jean-Pierre},
      TITLE = {Cohomological Invariants in {G}alois Cohomology},
      SERIES = {University Lecture Series},
      NUMBER = {28},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2003},
      PAGES = {viii+168},
      ISBN = {0-8218-3287-5},
      MRCLASS = {11E72 (12G05)},
      MRNUMBER = {2004f:11034},
      MRREVIEWER = {Gr{é}gory Berhuy},
      ZBLNUMBER = {1159.12311},
      }
  • [km2] Go to document N. A. Karpenko and A. S. Merkurjev, "Essential dimension of finite $p$-groups," Invent. Math., vol. 172, iss. 3, pp. 491-508, 2008.
    @article {km2, MRKEY = {2393078},
      AUTHOR = {Karpenko, Nikita A. and Merkurjev, Alexander S.},
      TITLE = {Essential dimension of finite {$p$}-groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {172},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {491--508},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {12F10 (14C35 16K20)},
      MRNUMBER = {2009b:12009},
      MRREVIEWER = {Zinovy Reichstein},
      DOI = {10.1007/s00222-007-0106-6},
      ZBLNUMBER = {05279042},
      }
  • [invol] M. Knus, A. Merkurjev, M. Rost, and J. Tignol, The Book of Involutions, Providence, RI: Amer. Math. Soc., 1998.
    @book {invol, MRKEY = {1632779},
      AUTHOR = {Knus, Max-Albert and Merkurjev, Alexander and Rost, Markus and Tignol, Jean-Pierre},
      TITLE = {The Book of Involutions},
      SERIES = {Amer. Math. Soc. Colloq. Publ.},
      NUMBER = {44},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1998},
      PAGES = {xxii+593},
      ISBN = {0-8218-0904-0},
      MRCLASS = {16K20 (11E39 11E57 11E72 11E88 16W10 20G10)},
      MRNUMBER = {2000a:16031},
      MRREVIEWER = {A. R. Wadsworth},
      ZBLNUMBER = {0955.16001},
      }
  • [lam] T. Y. Lam, The Algebraic Theory of Quadratic Forms, W. A. Benjamin, Reading, MA, 1973.
    @book {lam, MRKEY = {0396410},
      AUTHOR = {Lam, T. Y.},
      TITLE = {The Algebraic Theory of Quadratic Forms},
      NOTE = {Mathematics Lecture Note Series},
      PUBLISHER = {W. A. Benjamin, Reading, MA},
      YEAR = {1973},
      PAGES = {xi+344},
      MRCLASS = {10C05 (15A63)},
      MRNUMBER = {53 \#277},
      MRREVIEWER = {J. S. Hsia},
      ZBLNUMBER = {0259.10019},
      }
  • [meohni] A. Merkurjev, "Essential dimension," in Quadratic Forms — Algebra, Arithmetic , and Geometry, , 2009, vol. 493, pp. 299-326.
    @incollection{meohni,
      author={Merkurjev, A.},
      TITLE={Essential dimension},
      BOOKTITLE={Quadratic Forms --- Algebra, Arithmetic , and Geometry},
      NOTE={(R. Baeza, W. K. Chan, D. W. Hoffman, and R. Schulze-Pillot, eds.)},
      SERIES={Contemp. Math.},
      VOLUME={493},
      YEAR={2009},
      PAGES={299--326},
      }
  • [mrohni] A. Meyer and Z. Reichstein, Some consequences of the Karpenko-Merkurjev theorem.
    @misc{mrohni,
      author={Meyer, A. and Reichstein, Z.},
      TITLE={Some consequences of the Karpenko-Merkurjev theorem},
      NOTE={{\it Documenta Math.},
      to appear},
      ARXIV={0811.2517},
      }
  • [pstohni] R. Parimala, V. Suresh, and J. -P. Tignol, "On the Pfister number of quadratic forms," in Quadratic Forms — Algebra, Arithmetic, and Geometry, Baeza, R., Chan, W. K., Hoffman, D. W., and Schulze-Pillot, R., Eds., , 2009, vol. 493, pp. 327-338.
    @incollection{pstohni,
      author={Parimala, R. and Suresh, V. and Tignol, J.-P.},
      TITLE={On the Pfister number of quadratic forms},
      BOOKTITLE={Quadratic Forms --- Algebra, Arithmetic, and Geometry},
      EDITOR={Baeza, R. and Chan, W. K. and Hoffman, D. W. and Schulze-Pillot, R.},
      SERIES={Contemp. Math.},
      VOLUME={493},
      YEAR={2009},
      PAGES={327--338},
      }
  • [ampopov] A. M. Popov, "Finite stationary subgroups in general position of simple linear Lie groups," Trudy Moskov. Mat. Obshch., vol. 48, pp. 7-59, 263, 1985.
    @article {ampopov, MRKEY = {830410},
      AUTHOR = {Popov, A. M.},
      TITLE = {Finite stationary subgroups in general position of simple linear {L}ie groups},
      JOURNAL = {Trudy Moskov. Mat. Obshch.},
      FJOURNAL = {Trudy Moskovskogo Matematicheskogo Obshchestva},
      VOLUME = {48},
      YEAR = {1985},
      PAGES = {7--59, 263},
      ISSN = {0134-8663},
      MRCLASS = {22E10 (22E46)},
      MRNUMBER = {87i:22021},
      MRREVIEWER = {H. de Vries},
      ZBLNUMBER={0661.22009},
      }
  • [pv] V. L. Popov and E. B. Vinberg, Algebraic Geometry. IV, New York: Springer-Verlag, 1994.
    @book {pv, MRKEY = {1309681},
      AUTHOR={Popov, V. L and Vinberg, E. B.},
      TITLE = {Algebraic Geometry. {\rm IV}},
      SERIES = {Encyclopaedia of Mathematical Sciences},
      NUMBER = {55},
      NOTE = { A translation of {\it Algebraic Geometry} {\bf 4} (Russian), Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [MR1100483 (91k:14001)]},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1994},
      PAGES = {123--184},
      ISBN = {3-540-54682-0},
      MRCLASS = {14-06 (20-06)},
      MRNUMBER = {95g:14002},
      }
  • [reichstein] Go to document Z. Reichstein, "On the notion of essential dimension for algebraic groups," Transform. Groups, vol. 5, iss. 3, pp. 265-304, 2000.
    @article {reichstein, MRKEY = {1780933},
      AUTHOR = {Reichstein, Z.},
      TITLE = {On the notion of essential dimension for algebraic groups},
      JOURNAL = {Transform. Groups},
      FJOURNAL = {Transformation Groups},
      VOLUME = {5},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {265--304},
      ISSN = {1083-4362},
      MRCLASS = {20G15 (14L30)},
      MRNUMBER = {2001j:20073},
      MRREVIEWER = {Jorge A. Vargas},
      DOI = {10.1007/BF01679716},
      ZBLNUMBER = {0981.20033},
      }
  • [richardson] Go to document R. W. Richardson, "Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups," Duke Math. J., vol. 57, iss. 1, pp. 1-35, 1988.
    @article {richardson, MRKEY = {952224},
      AUTHOR = {Richardson, R. W.},
      TITLE = {Conjugacy classes of {$n$}-tuples in {L}ie algebras and algebraic groups},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {57},
      YEAR = {1988},
      NUMBER = {1},
      PAGES = {1--35},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {20G15 (14L30 17B45 22E46)},
      MRNUMBER = {89h:20061},
      MRREVIEWER = {Michel Brion},
      DOI = {10.1215/S0012-7094-88-05701-8},
      ZBLNUMBER = {0685.20035},
      }
  • [rost] M. Rost, "A descent property for Pfister forms," J. Ramanujan Math. Soc., vol. 14, iss. 1, pp. 55-63, 1999.
    @article {rost, MRKEY = {1700870},
      AUTHOR = {Rost, Markus},
      TITLE = {A descent property for {P}fister forms},
      JOURNAL = {J. Ramanujan Math. Soc.},
      FJOURNAL = {Journal of the Ramanujan Mathematical Society},
      VOLUME = {14},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {55--63},
      ISSN = {0970-1249},
      MRCLASS = {11E81 (19D45)},
      MRNUMBER = {2000f:11043},
      MRREVIEWER = {Jean-Pierre Tignol},
      ZBLNUMBER = {1059.11033},
      }
  • [ry] Z. Reichstein and B. Youssin, "Essential dimensions of algebraic groups and a resolution theorem for $G$-varieties," Canad. J. Math., vol. 52, iss. 5, pp. 1018-1056, 2000.
    @article {ry, MRKEY = {1782331},
      AUTHOR = {Reichstein, Zinovy and Youssin, Boris},
      TITLE = {Essential dimensions of algebraic groups and a resolution theorem for {$G$}-varieties},
      NOTE = {With an appendix by J{á}nos Koll{á}r and Endre Szab{ó}},
      JOURNAL = {Canad. J. Math.},
      FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      VOLUME = {52},
      YEAR = {2000},
      NUMBER = {5},
      PAGES = {1018--1056},
      ISSN = {0008-414X},
      CODEN = {CJMAAB},
      MRCLASS = {14L30 (14E15 20G15)},
      MRNUMBER = {2001k:14088},
      MRREVIEWER = {Lucy Moser-Jauslin},
      ZBLNUMBER = {1044.14023},
      }
  • [wood] Go to document J. A. Wood, "Spinor groups and algebraic coding theory," J. Combin. Theory Ser. A, vol. 51, iss. 2, pp. 277-313, 1989.
    @article {wood, MRKEY = {1001269},
      AUTHOR = {Wood, Jay A.},
      TITLE = {Spinor groups and algebraic coding theory},
      JOURNAL = {J. Combin. Theory Ser. A},
      FJOURNAL = {Journal of Combinatorial Theory. Series A},
      VOLUME = {51},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {277--313},
      ISSN = {0097-3165},
      CODEN = {JCBTA7},
      MRCLASS = {20F38 (94B25)},
      MRNUMBER = {90i:20040},
      MRREVIEWER = {Vera Pless},
      DOI = {10.1016/0097-3165(89)90053-8},
      ZBLNUMBER = {0704.22010},
      }

Authors

Patrick Brosnan

University of British Columbia, Department of Mathematics, 1984 Mathematics Road, Vancouver, V6T1Z2, Canada

Zinovy Reichstein

University of British Columbia, Department of Mathematics, 1984 Mathematics Road, Vancouver, V6T1Z2, Canada

Angelo Vistoli

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy