Book contents
- Frontmatter
- Contents
- Introduction
- 1 Finite group schemes
- 2 Algorithms for polycyclic groups
- 3 The spread of finite and infinite groups
- 4 Discrete subgroups of semisimple Lie groups, beyond lattices
- 5 Complete reducibility and subgroups of exceptional algebraic groups
- 6 Axial algebras of Jordan and Monster type
- 7 An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams
- 8 Finite groups and the class-size prime graph revisited
- 9 Character bounds for finite simple groups and applications
- 10 Generalized Baumslag-Solitar groups: a topological approach
- References
1 - Finite group schemes
Published online by Cambridge University Press: 21 November 2024
Book contents
- Frontmatter
- Contents
- Introduction
- 1 Finite group schemes
- 2 Algorithms for polycyclic groups
- 3 The spread of finite and infinite groups
- 4 Discrete subgroups of semisimple Lie groups, beyond lattices
- 5 Complete reducibility and subgroups of exceptional algebraic groups
- 6 Axial algebras of Jordan and Monster type
- 7 An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams
- 8 Finite groups and the class-size prime graph revisited
- 9 Character bounds for finite simple groups and applications
- 10 Generalized Baumslag-Solitar groups: a topological approach
- References
Summary
These extended notes give an introduction to the theory of finite group schemes over an algebraically closed field, with minimal prerequisites. They conclude with a brief survey of the inverse Galois problem for automorphism group schemes.
Keywords
- Type
- Chapter
- Information
- Groups St Andrews 2022 in Newcastle , pp. 1 - 52Publisher: Cambridge University PressPrint publication year: 2024
References
[1] Blanc, J., Brion, M., “Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics,” Ann. Fac. Sci. Toulouse Math. (6) 32 (2023), 607–622.CrossRefGoogle Scholar
[2] Bragg, D., “Automorphism groups of curves over arbitrary fields,” preprint, https://arxiv.org/abs/2304.02778Google Scholar
[3] Brion, M., “Automorphism groups of almost homogeneous varieties,” in: Facets of algebraic geometry. A collection in honor of William Fulton’s 80th birthday. Volume 1, London Math. Soc. Lecture Note Series 472, pp. 54–76, Cambridge Univ. Press, 2022.Google Scholar
[4] Brion, M., Schröer, S., “The inverse Galois problem for connected algebraic groups,” preprint, https://arxiv.org/abs/2205.08117Google Scholar
[5] Cushing, D., Stagg, G., Stewart, D., “A Prolog assisted search for new simple Lie algebras,” Math. Comp. 93 (2024), 1473–1495.Google Scholar
[6] Darda, R., Yasuda, T., “Inverse Galois problem for semicommutative finite group schemes,” preprint, https://arxiv.org/abs/2210.01495Google Scholar
[8] Demazure, M., Grothendieck, A., “Séminaire de Géométrie Algébrique du Bois Marie, 1962–64, Schémas en groupes (SGA3),” Tome I. Propriétés générales des schémas en groupes, Doc. Math. 7, Soc. Math. France, Paris, 2011.Google Scholar
[9] Dinh, T., Oguiso, K., “A surface with discrete and nonfinitely generated automorphism group,” Duke Math. J. 168 (2019), 941–966.CrossRefGoogle Scholar
[10] Eisenbud, D., Harris, J., “The geometry of schemes,” Graduate Texts Math. 197, Springer, New York, 2000.Google Scholar
[11] Eisenbud, D., “Commutative algebra with a view towards algebraic geometry,” Graduate Texts Math. 150, Springer, New York, 1996.Google Scholar
[12] Florence, M., “Realization of Abelian varieties as automorphism groups,” preprint, https://arxiv.org/abs/2102.02581Google Scholar
[13] Greenberg, L., “Maximal groups and signatures,” in: Discontinuous groups and Riemann surfaces, pp. 207–226. Princeton Univ. Press, Princeton, N.J., 1974.CrossRefGoogle Scholar
[14] Grothendieck, A., “Techniques de construction et théorèmes déxistence en géométrie algébrique IV: les schémas de Hilbert,” Sém. Bourbaki, Vol. 6 (1960–1961), Exp. 221, 249–276.Google Scholar
[15] Hartshorne, R., “Algebraic geometry,” Graduate Texts Math. 52, Springer, New York, 1977.Google Scholar
[16] Lesieutre, J., “A projective variety with discrete, non-finitely generated automorphism group,” Invent. Math. 212 (2018), 189–211.Google Scholar
[17] Liedtke, C., “A McKay correspondence in positive characteristic,” preprint, https://arxiv.org/abs/2207.06286Google Scholar
[18] Lombardo, D., Maffei, A., “Abelian varieties as automorphism groups of smooth projective varieties,” Int. Math. Res. Not. (2020), 1942—1956.CrossRefGoogle Scholar
[19] Madan, M., Rosen, M., “The group of automorphisms of a function field,” Proc. Amer. Math. Soc. 115 (1992), 923–929.CrossRefGoogle Scholar
[20] Madden, D., Valentini, R., “The group of automorphisms of algebraic function fields,” J. Reine Angew. Math. 343 (1983), 162–168.Google Scholar
[21] Malle, G., Matzat, B., “Inverse Galois theory. 2nd edition,” Springer Monographs in Mathematics, Springer, Berlin, 2018.CrossRefGoogle Scholar
[22] Martin, G., “Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces,” Algebra Number Theory 16 (2022), 1655–1704.CrossRefGoogle Scholar
[23] Martin, G., “Automorphism group schemes of bielliptic and quasibielliptic surfaces,” Épiother Géom. Algébrique 6 (2022), Article no. 9.Google Scholar
[24] Matsumura, H., Oort, F., “Representability of group functors, and automorphisms of algebraic schemes,” Invent. Math. 4 (1967), 1–25.CrossRefGoogle Scholar
[25] Milne, J. S., “Algebraic groups. The theory of group schemes of finite type over a field,” Cambridge Stud. Adv. Math. 170, Cambridge Univ. Press, Cambridge, 2017.Google Scholar
[27] Neukirch, J., Schmidt, A., Wingberg, K., “Cohomology of number fields. 2nd ed.,” Grundlehren Math. Wiss. 323, Springer, Berlin, 2008.Google Scholar
[28] Oguiso, K., “A surface in odd characteristic with discrete and non-finitely generated automorphism group,” Adv. Math. 375 (2020) 107397, 20 pp.CrossRefGoogle Scholar
[29] Pink, R., “Finite group schemes,” course notes available at https://people.math.ethz.ch/˜pink/ftp/FGS/CompleteNotes.pdfGoogle Scholar
[30] S. Schröer, Tziolas, N., “The structure of Frobenius kernels for automorphism group schemes,” Algebra Number Theory 17 (2023), 1637–1680,Google Scholar
[31] Serre, J.-P., “Topics in Galois theory. Notes written by Henri Darmon. 2nd ed.,” Research Notes in Math. 1, Wellesley, MA, 2007.Google Scholar
[32] Silverman, J. H., “The arithmetic of elliptic curves. Second edition,” Graduate Texts Math. 106, Springer, New York, 2009.Google Scholar
[33] Strade, H., “Simple Lie algebras over fields of positive characteristic. I, II, III,” de Gruyter Expositions in Mathematics 38, 42, 57, de Gruyter, Berlin, 2013–2017.Google Scholar
[34] Tate, J., “Finite flat group schemes,” in: Modular forms and Fermat’s last theorem, pp. 121–154, Springer, New York, 1997.CrossRefGoogle Scholar
[35] Tziolas, N., “Automorphisms of smooth canonically polarized surfaces in positive characteristic,” Adv. Math. 310 (2017), 235–289; corrigendum ibid., 585–593.CrossRefGoogle Scholar
[36] Viviani, F., “ Simple finite group schemes and their infinitesimal deformations,” Rend. Sem. Mat. Univ. Politec. Torino 68 (2010), 171–182.Google Scholar
[37] Waterhouse, W., “Introduction to affine group schemes,” Graduate Texts Math. 66, Springer, New York, 1979.Google Scholar