Algebra & Number Theory

Vinberg's representations and arithmetic invariant theory

Jack Thorne

Full-text: Open access

Abstract

Recently, Bhargava and others have proved very striking results about the average size of Selmer groups of Jacobians of algebraic curves over as these curves are varied through certain natural families. Their methods center around the idea of counting integral points in coregular representations, whose rational orbits can be shown to be related to Galois cohomology classes for the Jacobians of these algebraic curves.

In this paper we construct for each simply laced Dynkin diagram a coregular representation (G,V) and a family of algebraic curves over the geometric quotient VG. We show that the arithmetic of the Jacobians of these curves is related to the arithmetic of the rational orbits of G. In the case of type A2, we recover the correspondence between orbits and Galois cohomology classes used by Birch and Swinnerton-Dyer and later by Bhargava and Shankar in their works concerning the 2-Selmer groups of elliptic curves over .

Article information

Source
Algebra Number Theory, Volume 7, Number 9 (2013), 2331-2368.

Dates
Received: 8 November 2012
Revised: 14 February 2013
Accepted: 17 March 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730095

Digital Object Identifier
doi:10.2140/ant.2013.7.2331

Mathematical Reviews number (MathSciNet)
MR3152016

Zentralblatt MATH identifier
1321.11045

Subjects
Primary: 20G30: Linear algebraic groups over global fields and their integers
Secondary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10]

Keywords
arithmetic invariant theory Galois cohomology arithmetic of algebraic curves

Citation

Thorne, Jack. Vinberg's representations and arithmetic invariant theory. Algebra Number Theory 7 (2013), no. 9, 2331--2368. doi:10.2140/ant.2013.7.2331. https://projecteuclid.org/euclid.ant/1513730095


Export citation

References

  • V. I. Arnol'd, V. V. Goryunov, O. V. Lyashko, and V. A. Vasil'ev, “\cyr Osobennosti, I: \cyr Pokal'naya i global'naya teoriya”, pp. 5–257 in Dynamical systems 6, Itogi Nauki i Tekhniki Sovrem. Probl. Mat. Fund. Napr. 6, VINITI, Moscow, 1988. Translated as “Singularities, I: Local and global theory” in Singularity theory I, Encycl. Math. Sci. 6, Springer, Berlin, 1998.
  • M. Bhargava and B. H. Gross, “The average size of the $2$-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point”, preprint, 2013, http://www.math.harvard.edu/~gross/preprints/stable23.pdf.
  • M. Bhargava and W. Ho, “Coregular spaces and genus one curves”, preprint, 2013.
  • M. Bhargava and W. Ho, “On the average sizes of Selmer groups in families of elliptic curves”, In preparation.
  • M. Bhargava and A. Shankar, “Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves”, preprint, 2010. To appear in Annals of Math.
  • N. Bourbaki, Groupes et algèbres de Lie, chapitres 4–6, Actualités Scientifiques et Industrielles 1337, Hermann, Paris, 1968. Translated as Lie groups and Lie algebras, Chapters 4–6, Springer, Berlin, 2008.
  • N. Bourbaki, Groupes et algèbres de Lie, chapitres 7 et 8, Actualités Scientifiques et Industrielles 1364, Hermann, Paris, 1975. Translated in Lie groups and Lie algebras, Chapters 7–9, Springer, Berlin, 2008.
  • E. Brieskorn, “Singular elements of semi-simple algebraic groups”, pp. 279–284 in Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 2, Gauthier-Villars, Paris, 1971.
  • D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York, 1993.
  • P. R. Cook, “Compactified Jacobians and curves with simple singularities”, pp. 37–47 in Algebraic geometry (Catania, 1993 and Barcelona, 1994), edited by P. E. Newstead, Lecture Notes in Pure and Appl. Math. 200, Dekker, New York, 1998.
  • D. Ž. \DJoković, “The closure diagram for nilpotent orbits of the split real form of $E\sb 7$”, Represent. Theory 5 (2001), 284–316.
  • D. Ž. \DJoković and M. Litvinov, “The closure ordering of nilpotent orbits of the complex symmetric pair $({\rm SO}\sb {p+q},{\rm SO}\sb p\times{\rm SO}\sb q)$”, Canad. J. Math. 55:6 (2003), 1155–1190.
  • H. Esnault, “Sur l'identification de singularités apparaissant dans des groupes algébriques complexes”, pp. 31–59 in Seminar on Singularities (Paris, 1976–1977), edited by D. T. Lê, Publ. Math. Univ. Paris VII 7, Univ. Paris VII, Paris, 1980.
  • W. A. de Graaf, “Computing representatives of nilpotent orbits of $\theta$-groups”, J. Symbolic Comput. 46:4 (2011), 438–458.
  • B. H. Gross and M. Bhargava, “Arithmetic invariant theory”, preprint, 2012, http://www.math.harvard.edu/~gross/preprints/invariant.pdf.
  • L. Gruson, S. V. Sam, and J. Weyman, “Moduli of abelian varieties, Vinberg $\theta$-groups, and free resolutions”, pp. 419–469 in Commutative algebra, edited by I. Peeva, Springer, New York, 2013.
  • W. Ho, Orbit parametrizations of curves, thesis, Princeton University, Princeton, NJ, 2009, http://tinyurl.com/hothesis.
  • J. E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics 21, Springer, New York, 1975.
  • N. Kawanaka, “Orbits and stabilizers of nilpotent elements of a graded semisimple Lie algebra”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34:3 (1987), 573–597.
  • B. Kostant and S. Rallis, “Orbits and representations associated with symmetric spaces”, Amer. J. Math. 93 (1971), 753–809.
  • P. Levy, “Involutions of reductive Lie algebras in positive characteristic”, Adv. Math. 210:2 (2007), 505–559.
  • P. Levy, “Vinberg's $\theta$-groups in positive characteristic and Kostant–Weierstrass slices”, Transform. Groups 14:2 (2009), 417–461.
  • J. S. Milne, “Jacobian varieties”, pp. 167–212 in Arithmetic geometry (Storrs, CT, 1984), edited by G. Cornell and J. H. Silverman, Springer, New York, 1986.
  • A. L. Onishchik and È. B. Vinberg, \cyr Seminar po gruppam Li i algebraicheskim gruppam, Nauka, Moscow, 1988. Translated as Lie groups and algebraic groups, Springer, Berlin, 1990.
  • D. I. Panyushev, “On invariant theory of $\theta$-groups”, J. Algebra 283:2 (2005), 655–670.
  • B. Poonen, “Average rank of elliptic curves: after Manjul Bhargava and Arul Shankar”, pp. [exposé no.] 1049 in Séminaire Bourbaki, volume 2011/2012, Astérisque 352, Soc. Math. de France, Paris, 2013.
  • M. Reeder, “Elliptic centralizers in Weyl groups and their coinvariant representations”, Represent. Theory 15 (2011), 63–111.
  • M. Reeder, P. Levy, J.-K. Yu, and B. H. Gross, “Gradings of positive rank on simple Lie algebras”, Transform. Groups 17:4 (2012), 1123–1190.
  • J. Sekiguchi and Y. Shimizu, “Simple singularities and infinitesimally symmetric spaces”, Proc. Japan Acad. Ser. A Math. Sci. 57:1 (1981), 42–46.
  • N. I. Shepherd-Barron, “On simple groups and simple singularities”, Israel J. Math. 123 (2001), 179–188.
  • P. Slodowy, “Four lectures on simple groups and singularities”, Communications of the Mathematical Institute 11, Rijksuniversiteit Utrecht, 1980.
  • P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics 815, Springer, Berlin, 1980.
  • T. A. Springer, Invariant theory, Lecture Notes in Mathematics 585, Springer, Berlin, 1977.
  • T. A. Springer, Linear algebraic groups, Second ed., Birkhäuser, Boston, 2009.
  • R. Steinberg, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics 366, Springer, Berlin, 1974.
  • È. B. Vinberg, “\cyr Gruppa Veĭlya graduirovannoĭ algebry Li”, Izv. Akad. Nauk SSSR Ser. Mat. 40:3 (1976), 488–526. Translated as “The Weyl group of a graded Lie algebra” in Math. USSR-Izvestiya 10:3 (1976), 463–495.