Skip to main content
SpringerLink
Log in

Linear algebraic groups with good reduction

  • Research
  • Published:
Research in the Mathematical Sciences Aims and scope Submit manuscript

Abstract

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but now it appears to be developing into one of the central topics in the emerging arithmetic theory of (linear) algebraic groups over higher-dimensional fields. The focus of this article is on the Main Conjecture (Conjecture 5.7) asserting the finiteness of the number of isomorphism classes of forms of a given reductive group over a finitely generated field that have good reduction at a divisorial set of places of the field. Various connections between this conjecture and other problems in the theory of algebraic groups (such as the analysis of the global-to-local map in Galois cohomology and the genus problem) are discussed in detail. The article also includes a brief review of the required facts about discrete valuations, forms of algebraic groups, and Galois cohomology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. More formally, let \(v_p\) denote the (normalized) p-adic valuation on \({{\mathbb {Q}}}\) and let \({{\mathbb {Z}}}_{(p)}\) be the corresponding valuation ring. The elliptic curve E is said to have good reduction at p if there exists an abelian scheme \(E_{(p)}\) over the valuation ring \({{\mathbb {Z}}}_{(p)}\) with generic fiber E (the scheme \(E_{(p)}\) is then unique, which leads to a well-defined notion of reduction modulo p).

  2. We recall that one defines the unipotent radical of a connected algebraic group G to be the largest connected unipotent normal subgroup, and one says that G is reductive if its unipotent radical is trivial. For example, all tori (i.e., connected diagonalizable algebraic groups) are reductive. An algebraic group is (absolutely almost) simple if it does not contain any proper connected normal subgroups, and semi-simple if it admits a surjective morphism from a direct product of simple groups. We refer the reader to [10] and [32] for the details.

  3. In more technical terms, this system defines a scheme over \({{\mathbb {Z}}}_{(p)}\) with generic fiber G.

  4. For comparison, we would like to point out the following finiteness theorem for the forms of abelian varieties (cf. [89, §3, Finiteness theorem for forms]): Let X be an abelian variety over a field K, and let F/K be a finite separable extension. Then the set of K-isomorphism classes of abelian K-varieties \(X'\)such that there exists an F-isomorphism \(X \times _K F \simeq X' \times _K F\)is finite. On the contrary, for a semi-simple linear algebraic K-group G, and a finite separable extension F/K, the set of K-isomorphism classes of F/K-forms \(G'\) of G is infinite in many cases, even when K is a number field (see, however, the discussion of fields of type (F) in Sect. 5.2). So, the problem of classifying forms of (semi-simple) linear algebraic groups with special properties, which is central to the current article, comprises some challenges that do not arise in the context of abelian varieties.

  5. As defined in [32, §5].

  6. Using twisting, one shows that the properness of \(\theta _{\mathrm {PSL}_n , V}\) in fact implies the properness of \(\theta _{\mathrm {PSL}_{1 , A} , V}\) for any central simple K-algebra A of degree n.

  7. We observe that if the genus \(\mathbf{gen }(D)\) is infinite for a central division K-algebra D, then the genus \(\mathbf{gen }_K(G)\) is also infinite for the corresponding algebraic group \(G = \mathrm {SL}_{1 , D}\).

  8. We will not go into the details of this analysis here and would only like to point out that one of the important factors is the existence of so-called generic elements in every Zariski-dense subgroup—see [99] for a detailed discussion. The reader interested in the technical ingredients can also review the Isogeny Theorem (Theorem 4.2) in [96], which provides a far-reaching generalization of the following fact used in section Sect. 8.3: if \(\gamma _1 , \gamma _2 \in \mathrm {SL}_2(F)\)are semi-simple elements of infinite order that are weakly commensurable, then for any subfield K that contains the traces of \(\gamma _1\)and \(\gamma _2\), the subalgebras \(K[\gamma _1]\)and \(K[\gamma _2]\)are K-isomorphic.

  9. This means that that there exists an F-isomorphism \(\varphi :{\overline{G}}_1 \rightarrow {\overline{G}}_2\) between the corresponding adjoint groups such that \(\varphi ({\overline{\Gamma }}_1)\) is commensurable with \({\overline{\Gamma }}_2\), where \({\overline{\Gamma }}_i\) denotes the image of \(\Gamma _i\) in \({\overline{G}}_i(F)\).

  10. See Sect. 10.2 for some rigidity results over rings more general than rings of algebraic S-integers

  11. Of course, the traces of elements in the adjoint representation that generate the field of definition can be easily expressed in terms of the eigenvalues, but in our set-up, all we can work with are relations like (WC) in Definition 9.1 for \(\gamma _1 \in \Gamma _1\) and \(\gamma _2 \in \Gamma _2\), which do not immediately yield any relation between \(\mathrm {tr}(\mathrm {Ad}\, \gamma _1)\) and \(\mathrm {tr}(\mathrm {Ad}\, \gamma _2)\).

  12. Let us point out that here we deviate from the standard terminology. Namely, recall that in the classical setting where K is a number field, \(V^K\) is the set of all places of K, and \(S \subset V^K\) is a finite subset with complement \(V = V^K \setminus S\), we say that G has strong approximation with respect to S if the diagonal embedding \(G(K) \hookrightarrow G({\mathbb {A}}(K , V))\) has dense image (cf. [94, Ch. 7]).

  13. We will say that \((\Phi , R)\) is a nice pair if 2 is a unit in R whenever \(\Phi \) contains a subsystem of type \({\mathsf {B}}_2\), and 2 and 3 are units in R whenever \(\Phi \) is of type \({\mathsf {G}}_2\).

References

  1. Abrashkin, V.A.: \(p\)-divisible groups over \({{{\mathbb{Z}}}}\). Izv. Akad. Nauk SSSR Ser. Mat. 41(5), 987–1007, 1199 (1977)

    MathSciNet  MATH  Google Scholar 

  2. Algebraic Number Theory, ed. by J.W.S. Cassels and A. Fröhlich, 2nd edn. London Mathematical Society (2010)

  3. Amitsur, S.A.: Generic splitting fields of central simple algebras. Ann. Math. 62, 8–43 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atiyah, M.F., MacDonald, I.G.: Introduction to Commutative Algebra. Westview Press, Boulder (1969)

    MATH  Google Scholar 

  5. Bass, H.: \(K\)-theory and stable algebra. Publ. Math. IHES 22, 5–60 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bass, H.: Some problems. In: “classical” algebraic \(K\)-theory, Algebraic \(K\)-theory, II: “Classical” algebraic \(K\)-theory and connections with arithmetic (Proceedings of the Conference in Battelle Memorial Institute, Seattle, Washinton, 1972). Lecture Notes in Mathematics, vol. 342, pp. 3–73. Springer, Berlin (1973)

  7. Bass, H., Milnor, J., Serre, J.P.: Solution of the congruence subgroup problem for \(SL_ n(n \ge 3)\) and \(Sp_{2n} (n \ge 2)\). Publ. Math. Inst. Hautes Etud. Sci. 33, 59–137 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beli, C., Gille, P., Lee, T.-Y.: Examples of algebraic groups of type \({{\sf G}}_2\) having the same maximal tori. Proc. Steklov Inst. Math. 292(1), 10–19 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Borel, A.: Properties and linear representations of Chevalley groups. In: Seminar on Algebraic Groups and Related Finite Groups. Lecture Notes in Mathematics, vol. 131, pp. 1–55. Springer (1970)

  10. Borel, A.: Linear Algebraic Groups, vol. 126, 2nd edn. Springer, New York (1997)

    MATH  Google Scholar 

  11. Borel, A.: Some finiteness properties of adele groups over number fields. Inst. Hautes Études Sci. Publ. Math. 16, 5–30 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  12. Borel, A., Serre, J.-P.: Théorèmes de finitude en cohomologie galoisienne. Comment. Math. Helv. 39, 111–164 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  13. Borel, A., Tits, J.: Homomorphismes ”abstraits” de groupes algébriques simples. Ann. Math. 97(3), 499–571 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. Boyarchenko, M., Rapinchuk, I.A.: On abstract representations of the groups of rational points of algebraic groups in positive characteristic. Arch. Math. (Basel) 107(6), 569–580 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bourbaki, N.: Commutative Algebra, vol. 1-7. Springer, New York (1989)

    MATH  Google Scholar 

  16. Chernousov, V.I., Gille, P., Pianzola, A.: Torsors over the punctured affine line. Am. J. Math. 134(6), 1541–1583 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chernousov, V.I., Neher, E., Pianzola, A., Yahorau, U.: On conjugacy of Cartan subalgebras in extended affine Lie algebras. Adv. Math. 290, 260–292 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: The genus of a division algebra and the unramified Brauer group. Bull. Math. Sci. 3, 211–240 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: Division algebras with the same maximal subfields. Russ. Math. Surv. 70(1), 91–122 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: On the size of the genus of a division algebra. Proc. Steklov Inst. Math. 292(1), 63–93 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: On some finiteness properties of algebraic groups over finitely generated fields. C. R. Acad. Sci. Paris Ser. I 354, 869–873 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: Spinor groups with good reduction. Compos. Math. 155, 484–527 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: The finiteness of the genus of a finite-dimensional division algebra, and generalizations. Israel J. Math. (to appear)

  24. Chernousov, V.I., Rapinchuk, A.S., Rapinchuk, I.A.: Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction (in preparation)

  25. Colliot-Thélène, J.-L.: Birational invariants, purity and the Gersten conjecture, in \(K\)-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992). In: Proceedings of Symposia in Pure Mathematics, vol. 58, Part 1, pp. 1–64. American Mathematical Society, Providence (1995)

  26. Colliot-Thélène, J.-L.: Quelques résultats de finitude pour le groupe \(SK_1\) d’une algèbre de biquaternions. K-Theory 10(1), 31–48 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Colliot-Thélène, J.-L., Parimala, R., Suresh, V.: Patching and local-global principles for homogeneous spaces over function fields of p-adic curves. Comment. Math. Helv. 87(4), 1011–1033 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Colliot-Thélène, J.-L., Sansuc, J.-J.: Fibrés quadratiques et composantes connexes réelles. Math. Ann. 244(2), 105–134 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. Conrad, B.: Finiteness theorems for algebraic groups over function fields. Compos. Math. 148(2), 555–639 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Conrad, B.: Non-split reductive groups over \({{{\mathbb{Z}}}}\), Autour des schémas en groupes II. In: Panor. Synthèses, vol. 46, pp. 193–253. Societe Mathematique de France, Paris (2015)

  31. Conrad, B.: Reductive group schemes, Autour des schémas en groupes, École d’été “Schémas en groupes,” vol. I (Luminy 2011). Societe Mathematique de France, Paris (2014)

  32. Conrad, B., Gabber, O., Prasad, G.: Pseudo-Reductive Groups. New Mathematical Monographs, 2nd edn. Cambridge University Press, Cambridge (2015)

    Book  MATH  Google Scholar 

  33. Cutkosky, S.D.: Introduction to Algebraic Geometry, GSM, vol. 188. AMS, New York (2018)

    Book  MATH  Google Scholar 

  34. Darmon, H.: Rational points on curves. Arithmet. Geomet. Clay Math. Proc. 8, 7–53 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Demazure, M., Michel, G.P.: Algébriques, Groupes: Tome I: Géométrie algébrique, généralités, groupes commutatifs. North-Holland Publishing Co., Amsterdam (1970)

    MATH  Google Scholar 

  36. Demazure, M., Grothendieck, A.: Schémas en groupes, Séminaire de géométrie algébrique du Bois Marie 1962/64 (SGA 3). Lecture Notes in Mathematics, vol. 151-153. Springer, New York (1970)

    MATH  Google Scholar 

  37. Elman, R., Karpenko, N., Merkurjev, A.: The Algebraic and Geometric Theory of Quadratic Forms, vol. 56. Amer. Math. Soc. Colloq. Publ., Providence (2008)

    MATH  Google Scholar 

  38. Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73(3), 349–366 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  39. Farb, B., Dennis, R.K.: Noncommutative Algebra, GTM, vol. 144. Springer, New York (1993)

    Book  MATH  Google Scholar 

  40. Fontaine, J.-M.: Il n’y a pas de variété abélienne sur \({{{\mathbb{Z}}}}\). Invent. Math. 81(3), 515–538 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  41. Forster, O.: Lectures on Riemann surfaces, GTM, vol. 81. Springer, New York (1981)

    Book  Google Scholar 

  42. Garibaldi, S.: Outer automorphisms of algebraic groups and determining groups by their maximal tori. Michigan Math. J. 61(2), 227–237 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Garibaldi, S., Saltman, D.: Quaternion Algebras with the Same Subfields, Quadratic Forms, Linear Algebraic Groups, and Cohomology. Developmental Mathematics, vol. 225-238, p. 18. Springer, New York (2010)

    Google Scholar 

  44. Garibaldi, S., Rapinchuk, A.S.: Weakly commensurable \(S\)-arithmetic subgroups in almost simple algebraic groups of types \({{\sf B}}\) and \({{\sf C}}\). Algebra Number Theory 7(5), 1147–1178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gille, P.: Torseurs sur la droite affine. Transform. Groups 7(3), 231–245 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Gille, P., Pianzola, A.: Isotriviality and étale cohomology of Laurent polynomial rings. J. Pure Appl. Algebra 212(4), 780–800 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Gille, P.: Le problème de Kneser-Tits, Séminaire Bourbaki. Vol. 2007/2008, Astérisque No. 326 (2009), Exp. No. 983, vii, pp. 39–81 (2010)

  48. Gille, P., Szamuely, T.: Central Simple Algebras and Galois Cohomology. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  49. Gross, B.H.: Groups over \({{{\mathbb{Z}}}}\). Invent. math. 124(1–3), 263–279 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  50. Guo, N.: The Grothendieck-Serre Conjecture over Semilocal Dedekind Rings. arXiv:1902.02315

  51. Harada, S., Hiranouchi, T.: Smallness of fundamental groups for arithmetic schemes. J. Number Theory 129(11), 2702–2712 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Harbater, D., Hartmann, J., Krashen, D.: Applications of patching to quadratic forms and central simple algebras. Invent. Math. 178(2), 231–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. Harbater, D., Hartmann, J., Krashen, D.: Local-global principles for torsors over arithmetic curves. Am. J. Math. 137(6), 1559–1612 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Harder, G.: Halbeinfache Gruppenschemata über Dedekindringen. Invent. math. 4, 165–191 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  55. Harder, G.: Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III. J. Reine Angew. Math. 274(5), 125–138 (1975)

    MathSciNet  MATH  Google Scholar 

  56. Hartshorne, R.: Algebraic Geometry, GTM, vol. 52. Springer, New York (1977)

    Book  MATH  Google Scholar 

  57. Izhboldin, O.T.: Motivic equivalence of quadratic forms. Doc. Math. 3, 341–351 (1998)

    MathSciNet  MATH  Google Scholar 

  58. Jannsen, U.: Principe de Hasse cohomologique. Séminaire de Théorie des Nombres, Paris 1989–90, pp. 121–140

  59. Jannsen, U.: Hasse principles for higher-dimensional fields. Ann. Math. 183(1), 1–71 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  60. Javanpeykar, A., Loughran, D.: Good reduction of algebraic groups and flag varieties. Arch. Math. 104(2), 133–143 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  61. Kac, M.: Can One Hear the Shape of a Drum?, vol. 4, 73rd edn, pp. 1–23. American Mathematical Monthly, New York (1966)

    MATH  Google Scholar 

  62. Kahn, B.: Sur le groupe des clsses d’un schéma arithmétiques. Bull. Soc. Math. France 134(3), 395–415 (2006)

    Article  MathSciNet  Google Scholar 

  63. Karpenko, N.A.: Criteria of motivic equivalence for quadratic forms and central simple algebras. Math. Ann. 317(3), 585–611 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  64. Kato, K.: A Hasse principle for two dimensional global fields. J. Reine Angew. Math. 366, 142–183 (1986)

    MathSciNet  MATH  Google Scholar 

  65. Kneser, M.: Starke approximation in algebraischen Gruppen. I. J. Reine Angew. Math. 218, 190–203 (1965)

    MathSciNet  MATH  Google Scholar 

  66. Kneser, M.: Strong approximation. In: Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics, Boulder, CO, 1965), pp. 187–196, American Mathematical Society, Providence

  67. Knus, M.-A., Merkurjev, A., Rost, M., Tignol, J.-P.: The Book of Involutions, vol. 44. American Mathematical Society Colloquium Publications, New York (1998)

    MATH  Google Scholar 

  68. Knus, M.A., Ojanguren, M.: Théorie de la descente et algèbres d’Azumaya. Lecture Notes in Mathematics, vol. 389. Springer, New York (1974)

    Book  MATH  Google Scholar 

  69. Krashen, D., McKinnie, K.: Distinguishing division algebras by finite splitting fields. Manuscripta Math. 134(1–2), 171–182 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  70. Lam, T.Y.: Introduction to Quadratic Forms over Fields, GSM 67. AMS, New York (2005)

    Google Scholar 

  71. Lang, S.: Fundamentals of Diophantine Geometry. Springer, New York (1983)

    Book  MATH  Google Scholar 

  72. Maclachlan, C., Reid, A.: The Arithmetic of Hyperbolic 3-Manifolds, GTM, vol. 219. Springer, New York (2003)

    Book  MATH  Google Scholar 

  73. Margulis, G.A.: Cobounded subgroups in algebraic groups over local fields. Funkcional. Anal. i Prilozen. 11(2), 45–57, 95 (1977)

    MathSciNet  MATH  Google Scholar 

  74. Margulis, G.A.: Discrete Subgroups of Semisimple Lie Groups. Springer, New York (1991)

    Book  MATH  Google Scholar 

  75. McKean, H.P.: Selberg’s trace formula as applied to a compact Riemann surface. Commun. Pure Appl. Math. 25, 225–246 (1972)

    Article  MathSciNet  Google Scholar 

  76. Meyer, J.S.: A division algebra with infinite genus. Bull. Lond. Math. Soc. 46, 463–468 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  77. Milne, J.S.: Class Field Theory. https://www.jmilne.org/math/CourseNotes/cft.html

  78. Milne, J.S.: Etale Cohomology. Princeton University Press, Princeton (1980)

    Book  MATH  Google Scholar 

  79. Milnor, J.: Introduction to Algebraic \(K\)-Theory. Annals of Mathematics Studies, vol. 72. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  80. Mordell, L.J.: On the rational solutions of the indeterminate equations of the third and fourth degrees. Proc. Camb. Philos. Soc. 21, 179–192 (1922)

    MATH  Google Scholar 

  81. Moret-Bailly, L.: Pinceaux de variétés abéliennes. Astérisque, vol. 129 (1985)

  82. Mumford, D.: Abelian Varieties, vol. 5. Tata Institute of Fundamental Research Studies in Mathematics, Mumbai (1970)

    MATH  Google Scholar 

  83. Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften, vol. 322. Springer, New York (1999)

    Book  MATH  Google Scholar 

  84. Nisnevich, E.A.: Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind. C. R. Acad. Sci. Paris Ser. I Math. 299(1), 5–8 (1984)

    MathSciNet  MATH  Google Scholar 

  85. O’Meara, O.T.: Introduction to Quadratic Forms. Die Grundlehren der mathematischen Wissenschaften, vol. 117. Springer, New York (1963)

    Book  MATH  Google Scholar 

  86. Orlov, D., Vishik, A., Voevodsky, V.: An exact sequence for \(K_*^M/2\) with applications to quadratic forms. Ann. Math. 165(1), 1–13 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  87. Parimala, R.: A Hasse principle for quadratic forms over function fields. Bull. Am. Math. Soc. (N.S.) 51(3), 447–461 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  88. Parshin, A.N.: Minimal models of curves of genus 2, and homomorphisms of abelian varieties defined over a field of finite characteristic. Izv. Akad. Nauk SSSR Ser. Mat. 36, 67–109 (1972)

    MathSciNet  Google Scholar 

  89. Parshin, A.N., Zarhin, Y.G.: Finiteness problems in Diophantine geometry, English version of the Appendix to the Russian translation of S. Lang’s Fundamentals of Diophantine geometry. arXiv:0912.4325

  90. Pierce, R.S.: Associative Algebras, GTM, vol. 88. Springer, New York (1982)

    Book  Google Scholar 

  91. Platonov, V.P.: The problem of strong approximation and the Kneser-Tits hypothesis for algebraic groups. Izv. Akad. Nauk SSSR Ser. Mat. 33, 1211–1219 (1969)

    MathSciNet  Google Scholar 

  92. Platonov, V.P.: On the Tannaka-Artin problem. Dokl. Akad. Nauk SSSR 221(5), 1038–1041 (1975)

    MathSciNet  MATH  Google Scholar 

  93. Platonov, V.P.: Reduced \(K\)-theory and approximation in algebraic groups. Trudy Mat. Inst. Steklov. 142, 198–207 (1976)

    MathSciNet  Google Scholar 

  94. Platonov, V.P., Rapinchuk, A.S.: Algebraic Groups and Number Theory. Academic Press, Cambridge (1994)

    MATH  Google Scholar 

  95. Prasad, G.: Strong approximation for semi-simple groups over function fields. Ann. Math. 105(3), 553–572 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  96. Prasad, G., Rapinchuk, A.S.: Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publ. Math. IHES 109, 113–184 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  97. Prasad, G., Rapinchuk, A.S.: Local-global principles for embedding of fields with involution into simple algebras with involution. Comment. Math. Helv. 85(3), 583–645 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  98. Prasad, G., Rapinchuk, A.S.: On the fields generated by the lengths of closed geodesics in locally symmetric spaces. Geom. Dedicata. 172, 79–120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  99. Prasad, G., Rapinchuk, A.S.: Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces. In: Thin Groups and Superstrong Approximation, vol. 61, pp. 211–252. Cambridge University Press, Cambridge (2014)

  100. Prasad, G., Rapinchuk, A.S.: Weakly Commensurable Groups, with Applications to Differential Geometry in Handbook of Group Actions I. Advanced Lectures in Mathematics, vol. 31. International Press, Somerville (2015)

    Google Scholar 

  101. Raghunathan, M.S.: Discrete Subgroups of Lie Groups Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 68. Springer, New York (1972)

    Book  Google Scholar 

  102. Raghunathan, M.S., Ramanathan, A.: Principal bundles on the affine line. Proc. Indian Acad. Sci. (Math. Sci.) 93(2–3), 137–145 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  103. Rapinchuk, A.S.: Strong Approximation for Algebraic Groups in Thin Groups and Superstrong Approximation, vol. 61, pp. 269–298. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  104. Rapinchuk, A.S.: Towards the eigenvalue rigidity of Zariski-dense subgroups. In: Proceedings of ICM-2014 (Seoul), vol. II, pp. 247-*269

  105. Rapinchuk, A.S., Rapinchuk, I.A.: On division algebras having the same maximal subfields. Manuscr. Math. 132, 273–293 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  106. Rapinchuk, A.S., Rapinchuk, I.A.: Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields. arXiv:2002.01520

  107. Rapinchuk, I.A.: On linear representations of Chevalley groups over commutative rings. Proc. Lond. Math. Soc. 102(5), 951–983 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  108. Rapinchuk, I.A.: On abstract representations of the groups of rational points of algebraic groups and their deformations. Algebra Number Theory 7(7), 1685–1723 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  109. Rapinchuk, I.A.: On the character varieties of finitely generated groups. Math. Res. Lett. 22(2), 579–604 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  110. Rapinchuk, I.A.: Abstract homomorphisms of algebraic groups and applications. In: Handbook of Group Actions II. Advanced Lectures in Mathematics (ALM), vol. 32, pp. 397–447. International Press, Somerville (2015)

  111. Rapinchuk, I.A.: On abstract homomorphisms of Chevalley groups over the coordinate rings of affine curves. Transform. Gr. 24(4), 1241–1259 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  112. Rapinchuk, I.A.: A generalization of Serre’s condition \((\rm F)\) with applications to the finiteness of unramified cohomology. Math. Z. 291(1–2), 199–213 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  113. Reid, A.: Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds. Duke Math. J. 65(2), 215–228 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  114. Saltman, D.: Lectures on division algebras. In: CBMS Regional Conference Series, vol. 94, AMS (1999)

  115. Samuel, P.: Anneaux gradués factoriels et modules réflexifs. Bull. Soc. Math. France 92, 237–249 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  116. Serre, J.-P.: Abelian \(\ell \)-adic representations and elliptic curves. McGill University Lecture Notes Written with the Collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc. (1968)

  117. Serre, J.-P.: A Course in Arithmetic, GTM, vol. 7. Springer, New York (1973)

    Book  MATH  Google Scholar 

  118. Serre, J.-P.: Galois Cohomology. Springer, New York (1997)

    Book  MATH  Google Scholar 

  119. Serre, J.-P.: Local Algebra. Springer, New York (2000)

    Book  MATH  Google Scholar 

  120. Serre, J.-P.: Le problème des groupes de congruence pour \(SL_2\). Ann. Math. 92, 489–527 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  121. Shafarevich, I.R.: Algebraic number fields. In: Proceedings of ICM-1962 (Stockholm), pp. 163–176

  122. Shatz, S.: Profinite Groups, Arithmetic, and Geometry. Annals of Mathematics Studies, vol. 67. Princeton University Press, Princeton (1972)

    Book  MATH  Google Scholar 

  123. Shenfeld, D.: On semisimple representations of universal lattices. Groups Geom. Dyn. 4(1), 179–193 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  124. Silverman, J.: The arithmetic of elliptic curves, GTM, vol. 106, 2nd edn. Springer, New York (2009)

    Book  Google Scholar 

  125. Silverman, J., Tate, J.: Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (2015)

    Book  MATH  Google Scholar 

  126. Springer, T.A.: Linear Algebraic Groups. Progress in Mathematics, vol. 9, 2nd edn. Birkhäuser Boston Inc., Boston (1998)

    Book  MATH  Google Scholar 

  127. Srinivasan, S.: Good Reduction of Unitary Groups of Quaternionic Skew-Hermitian Forms. arXiv:1906.01414

  128. Stavrova, A.: Chevalley groups of polynomial rings over Dedekind domains. J. Group Theory 23(1), 121–132 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  129. Steinberg, R.: Some consequence of elementary relations in \(SL_n\), finite groups—coming of age. In: Proceedings of the Canadian Mathematical Society Conference held on June 15–28, 1982. Contemporary Mathematics, vol. 45. AMS (1985)

  130. Suslin, A.A.: On the structure of the special linear group over rings of polynomials. Izv. Akad. Nauk SSSR Ser. Mat. 41, 235–252 (1977) [English transl. Math.-USSR Izv. 11, 221–238 (1977)]

  131. Szamuely, T.: Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, vol. 117. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  132. Szpiro, L.: Sur le théorème de rigidité de Parshin et Arakelov. Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II, pp. 169–202, Astérisque 64, Soc. Math. France, Paris (1979)

  133. Szpiro, L.: Propriétés numériques du faisceau dualisant relatif, seminar on pencils of curves of genus at least two. Astérisque 86, 44–78 (1981)

    MATH  Google Scholar 

  134. Thin groups and superstrong approximation, Selected expanded papers from the workshop held in Berkeley, CA, February 6–10, 2012, edited by E. Breuillard and H. Oh, Mathematical Sciences Research Institute Publications, vol. 61. Cambridge University Press, Cambridge (2014)

  135. Tikhonov, S.V.: Division algebras of prime degree with infinite genus. Proc. Steklov Inst. 292, 256–259 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  136. Tits, J.: Classification of algebraic semisimple groups. In: Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics, Boulder, CO, 1965, pp. 33–62, American Mathematical Society, Providence (1966)

  137. Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  138. Tits, J.: Groupes de Whitehead de groupes algébriques simples sur un corps (d’après V. P. Platonov et al.), Séminaire Bourbaki (1976/77), Exp. No. 505, pp. 218–236, Lecture Notes in Mathematics, vol. 677. Springer, New York (1978)

  139. Vinberg, E.B.: Rings of definition of dense subgroups of semisimple linear groups. Izv. Akad. Nauk SSSR Ser. Mat. 35, 45–55 (1971)

    MathSciNet  MATH  Google Scholar 

  140. Vishik, A.: Integral Motives of Quadrics, Preprint MPI-1998-13. Max Planck Institute für Mathematik, Bonn 1998, p. 82. http://www.mpim-bonn.mpg.de/node/263

  141. Vishik, A.: Motives of quadrics with applications to the theory of quadratic forms. Geometric methods in the algebraic theory of quadratic forms. Lecture Notes in Mathematics, vol. 1835, pp. 25–101. Springer, Berlin (2004)

  142. Voloch, J.F.: Diophantine geometry in characteristic \(p\): a survey. In: Arithmetic Geometry (Cortona, 1994), Symposium in Mathematics, vol. XXXVII, pp. 260–278. Cambridge University Press, Cambridge (1997)

  143. Voskresenskii, V.E.: Algebraic Groups and Their Birational Invariants. Translations of Mathematical Monographs, vol. 179. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  144. Waterhouse, W.: Introduction to Affine Group Schemes, GTM, vol. 66. Springer, New York (1979)

    Book  MATH  Google Scholar 

  145. Weisfeiler, B.: Semisimple algebraic groups that are decomposable over a quadratic extension. Izv. Akad. Nauk SSSR Ser. Mat. 35, 56–71 (1971)

    MathSciNet  Google Scholar 

  146. Yamasaki, A.: Strong approximation theorem for division algebras over \({{{\mathbb{R}}}}(X)\). J. Math. Soc. Jpn. 49(3), 455–467 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  147. Zarhin, Y.G.: Torsion of abelian varieties in finite characteristic. Mat. Zametki 22(1), 3–11 (1977)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

Special thanks are due to Brian Conrad, who carefully read the article and made a number of suggestions that helped to improve the exposition. We are also grateful to Uriya First, Ariyan Javanpeykar, Boris Kunyavskiī, Daniel Loughran, Alexander Merkurjev, Dipendra Prasad, C. Rajan, Zinovy Reichstein, Charlotte Ure, Uzi Vishne, and the anonymous referee for helpful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904-4137, USA

    Andrei S. Rapinchuk

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Igor A. Rapinchuk

Authors
  1. Andrei S. Rapinchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor A. Rapinchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor A. Rapinchuk.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rapinchuk, A.S., Rapinchuk, I.A. Linear algebraic groups with good reduction. Res Math Sci 7, 28 (2020). https://doi.org/10.1007/s40687-020-00226-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40687-020-00226-3

Search

Navigation