Algebra
Essential dimension of finite groups in prime characteristic
Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 463-467.

Let F be a field of characteristic p>0 and G be a smooth finite algebraic group over F. We compute the essential dimension edF(G;p) of G at p. That is, we show that

edF(G;p)={1,ifpdivides|G|,and0,otherwise.

Soit F un corps de caractéristique p>0, et soit G un groupe algébrique fini étale sur F. On calcule la dimension essentielle de G en p, que l'on note edF(G;p). Plus précisément, on démontre que

edF(G;p)={1,sipdivise|G|,0,sinon.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.03.013
Reichstein, Zinovy 1; Vistoli, Angelo 2

1 Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada
2 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
@article{CRMATH_2018__356_5_463_0,
     author = {Reichstein, Zinovy and Vistoli, Angelo},
     title = {Essential dimension of finite groups in prime characteristic},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {463--467},
     publisher = {Elsevier},
     volume = {356},
     number = {5},
     year = {2018},
     doi = {10.1016/j.crma.2018.03.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.03.013/}
}
TY  - JOUR
AU  - Reichstein, Zinovy
AU  - Vistoli, Angelo
TI  - Essential dimension of finite groups in prime characteristic
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 463
EP  - 467
VL  - 356
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.03.013/
DO  - 10.1016/j.crma.2018.03.013
LA  - en
ID  - CRMATH_2018__356_5_463_0
ER  - 
%0 Journal Article
%A Reichstein, Zinovy
%A Vistoli, Angelo
%T Essential dimension of finite groups in prime characteristic
%J Comptes Rendus. Mathématique
%D 2018
%P 463-467
%V 356
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.03.013/
%R 10.1016/j.crma.2018.03.013
%G en
%F CRMATH_2018__356_5_463_0
Reichstein, Zinovy; Vistoli, Angelo. Essential dimension of finite groups in prime characteristic. Comptes Rendus. Mathématique, Volume 356 (2018) no. 5, pp. 463-467. doi : 10.1016/j.crma.2018.03.013. http://www.numdam.org/articles/10.1016/j.crma.2018.03.013/

[1] Bleher, F.M.; Chinburg, T.; Poonen, B.; Symonds, P. Automorphisms of Harbater–Katz–Gabber curves, Math. Ann., Volume 368 (2017) no. 1–2, pp. 811-836 (MR3651589)

[2] Borel, A.; Springer, T.A. Rationality properties of linear algebraic groups. II, Tohoku Math. J. (2), Volume 20 (1968), pp. 443-497 (MR0244259)

[3] Camina, R. Subgroups of the Nottingham group, J. Algebra, Volume 196 (1997) no. 1, pp. 101-113 (MR1474165)

[4] Chernousov, V.; Gille, P.; Reichstein, Z. Resolving G-torsors by abelian base extensions, J. Algebra, Volume 296 (2006) no. 2, pp. 561-581 (MR2201056)

[5] Chernousov, V.; Gille, P.; Reichstein, Z. Reduction of structure for torsors over semilocal rings, Manuscr. Math., Volume 126 (2008) no. 4, pp. 465-480 (MR2425436)

[6] Duncan, A.; Reichstein, Z. Versality of algebraic group actions and rational points on twisted varieties, J. Algebraic Geom., Volume 24 (2015) no. 3, pp. 499-530 (MR3344763)

[7] Florence, M.; Reichstein, Z. The rationality problem for forms of moduli spaces of stable marked curves of positive genus | arXiv

[8] Grothendieck, A. Torsion homologique et sections rationnelles, Anneaux de Chow et Applications, Séminaire Claude-Chevalley, vol. 3, 1958, pp. 1-29 (exposé 5)

[9] Harbater, D. Moduli of p-covers of curves, Commun. Algebra, Volume 8 (1980) no. 12, pp. 1095-1122 (MR0579791)

[10] Karpenko, N.A.; Merkurjev, A.S. Essential dimension of finite p-groups, Invent. Math., Volume 172 (2008) no. 3, pp. 491-508 (MR2393078)

[11] Katz, N.M. Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble), Volume 36 (1986) no. 4, pp. 69-106 (MR0867916)

[12] Ledet, A. On the essential dimension of p-groups, Galois Theory and Modular Forms, Dev. Math., vol. 11, Kluwer Academic Publishers, Boston, MA, USA, 2004, pp. 159-172 (MR2059762)

[13] Ledet, A. Finite groups of essential dimension one, J. Algebra, Volume 311 (2007) no. 1, pp. 31-37 (MR2309876)

[14] Lötscher, R.; MacDonald, M.; Meyer, A.; Reichstein, Z. Essential p-dimension of algebraic groups whose connected component is a torus, Algebra Number Theory, Volume 7 (2013) no. 8, pp. 1817-1840 (MR3134035)

[15] Merkurjev, A.S. Essential dimension, Quadratic Forms—Algebra, Arithmetic, and Geometry, Contemp. Math., vol. 493, American Mathematical Society, Providence, RI, USA, 2009, pp. 299-325 (MR2537108)

[16] Merkurjev, A.S. Essential dimension: a survey, Transform. Groups, Volume 18 (2013) no. 2, pp. 415-481

[17] Meyer, A.; Reichstein, Z. The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra Number Theory, Volume 3 (2009) no. 4, pp. 467-487

[18] Meyer, A.; Reichstein, Z. Some consequences of the Karpenko–Merkurjev theorem, Doc. Math. (2010), pp. 445-457 (MR2804261)

[19] Reichstein, Z. Essential dimension, Proceedings of the International Congress of Mathematicians, Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 162-188

[20] Robinson, D.J.S. A Course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996 (MR1357169)

[21] Serre, J.-P. Galois Cohomology, Springer-Verlag, Berlin, 1997 (translated from the French by Patrick Ion and revised by the author MR1466966)

[22] Serre, J.-P. Sous-groupes finis des groupes de Lie, Astérisque, Volume 266 (2000), pp. 415-430 (Exp. No. 864, 5. MR1772682)

[23] Serre, J.-P. Cohomological invariants, Witt invariants, and trace forms, Cohomological Invariants in Galois Cohomology, Univ. Lecture Ser., vol. 28, American Mathematical Society, Providence, RI, USA, 2003, pp. 1-100 (Notes by Skip Garibaldi)

[24] Steinberg, R. Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math., Volume 25 (1965), pp. 49-80 (MR0180554)

[25] Tossici, D.; Vistoli, A. On the essential dimension of infinitesimal group schemes, Amer. J. Math., Volume 135 (2013) no. 1, pp. 103-114 (MR3022958)

Cited by Sources:

The authors are grateful to the Collaborative Research Group in Geometric and Cohomological Methods in Algebra at the Pacific Institute for the Mathematical Sciences for their support of this project.