Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-13T15:52:42.549Z Has data issue: false hasContentIssue false
  • English
  • Français

Finiteness theorems for algebraic groups over function fields

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  30 November 2011

Brian Conrad
Brian Conrad*
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (email: conrad@math.stanford.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

Keywords

class numbersTamagawa numbersTate–Shafarevich setspseudo-reductive groups

MSC classification

Secondary: 20G30: Linear algebraic groups over global fields and their integers 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 555 - 639
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Art69]Artin, M., Algebraization of formal moduli: I, in Global analysis (papers in honor of K. Kodaira) (University of Tokyo Press, Tokyo, 1969), 21–71.Google Scholar
[Bor98]Borcherds, R., Coxeter groups, Lorentzian lattices, and K3 surfaces, Int. Math. Res. Not. IMRN 19 (1998), 1011–1031.CrossRefGoogle Scholar
[Bor63]Borel, A., Some finiteness properties of adele groups over number fields, Publ. Math. Inst. Hautes Études Sci. 16 (1963), 5–30.CrossRefGoogle Scholar
[Bor91]Borel, A., Linear algebraic groups, second edition (Springer, New York, 1991).CrossRefGoogle Scholar
[BH62]Borel, A. and Harish-Chandra, , Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535.CrossRefGoogle Scholar
[BP90]Borel, A. and Prasad, G., Addendum to ‘Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups’, Publ. Math. Inst. Hautes Études Sci. 71 (1990), 173–177.CrossRefGoogle Scholar
[BS64]Borel, A. and Serre, J.-P., Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164.CrossRefGoogle Scholar
[BT65]Borel, A. and Tits, J., Groupes réductifs, Publ. Math. Inst. Hautes Études Sci. 27 (1965), 55–151.CrossRefGoogle Scholar
[BT73]Borel, A. and Tits, J., Homomorphismes ‘abstraits’ de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571.CrossRefGoogle Scholar
[BT78]Borel, A. and Tits, J., Théorèmes de structure et de conjugaison pour les groupes algébriques linéaires, C. R. Acad. Sci. Paris (A) 287 (1978), 55–57.Google Scholar
[BLR90]Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models (Springer, New York, 1990).CrossRefGoogle Scholar
[Bou98]Bourbaki, N., General topology (Springer, New York, 1998), Chapters 5–10.Google Scholar
[Bri09]Brion, M., Anti-affine algebraic groups, J. Algebra 321 (2009), 934–952.CrossRefGoogle Scholar
[Bro]Broshi, M., G-torsors over a Dedekind scheme, J. Pure Appl. Algebra, to appear.Google Scholar
[BrT84]Bruhat, F. and Tits, J., Groupes algébriques sur un corps local. Chapitre II. Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math. Inst. Hautes Études Sci. 60 (1984), 197–376.Google Scholar
[BrT87]Bruhat, F. and Tits, J., Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo Sect. IA 34 (1987), 671–698.Google Scholar
[Che60]Chevalley, C., Une démonstration d’un théorème sur les groupes algébriques, J. Math. Pures Appl. (9) 39 (1960), 307–317.Google Scholar
[CS04]Colmez, P. and Serre, J.-P., Grothendieck–Serre correspondence, bilingual edition (American Mathematical Society and Société Mathématique de France, 2004).Google Scholar
[Con02]Conrad, B., A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), 1–18.Google Scholar
[Con10]Conrad, B., Weil and Grothendieck approaches to adelic points, 2010 (submitted).Google Scholar
[CGP10]Conrad, B., Gabber, O. and Prasad, G., Pseudo–reductive groups (Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
[DG70]Demazure, M. and Gabriel, P., Groupes algébriques (Masson, Paris, 1970).Google Scholar
[SGA3]Demazure, M. and Grothendieck, A., Schémas en groupes I, II, III, Lecture Notes in Mathematics, vols 151, 152, 153 (Springer, New York, 1970).Google Scholar
[GM]Gille, P. and Moret-Bailly, L., Actions algébriques de groupes arithmétiques, in Torsors, theory and applications (Edinburgh, 2011), Proceedings of the London Mathematical Society, eds V. Batyrev and A. Skorobogatov (London Mathematical Society, London), to appear.Google Scholar
[EGA]Grothendieck, A., Eléments de Géométrie Algébrique, Publ. Math. Inst. Hautes Études Sci. 4, 8, 11, 17, 20, 24, 28, 32 (1960–67).Google Scholar
[Gro68]Grothendieck, A., Le groupe de Brauer III: exemples et compléments, in Dix Exposés sur la cohomologie des schémas (North-Holland, Amsterdam, 1968), 88–188.Google Scholar
[SGA1]Grothendieck, A., Revêtements étale et Groupe Fondamental, Lecture Notes in Mathematics, vol. 224 (Springer, New York, 1971).CrossRefGoogle Scholar
[Har69]Harder, G., Minkowskische Reduktionstheorie über Funktionenkörpern, Invent. Math. 7 (1969), 33–54.CrossRefGoogle Scholar
[Har75]Harder, G., Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, J. Reine Angew. Math. 274/5 (1975), 125–138.Google Scholar
[Lan96]Landvogt, E., A compactification of the Bruhat–Tits buildings, Springer Lecture Notes in Mathematics, vol. 1619 (Springer, New York, 1996).CrossRefGoogle Scholar
[Maz93]Mazur, B., On the passage from local to global in number theory, Bull. Amer. Math. Soc. 29 (1993), 14–50.CrossRefGoogle Scholar
[Mil80]Milne, J., Étale cohomology (Princeton University Press, 1980).Google Scholar
[Mil86]Milne, J., Arithmetic duality theorems (Academic Press, Boston, 1986).Google Scholar
[MT62]Mostow, G. and Tamagawa, T., On the compactness of arithmetically defined homogeneous spaces, Ann. of Math. (2) 76 (1962), 446–463.CrossRefGoogle Scholar
[NSW08]Neukirch, J., Schmidt, A. and Winberg, K., Cohomology of number fields, second edition (Springer, New York, 2008).CrossRefGoogle Scholar
[Nis82]Nisnevich, Y., Étale cohomology and arithmetic of semi-simple groups, PhD thesis, Harvard University (1982).Google Scholar
[Oes84]Oesterlé, J., Nombres de Tamagawa et groupes unipotents en caractéristique p, Invent. Math. 78 (1984), 13–88.CrossRefGoogle Scholar
[Pra77]Prasad, G., Strong approximation for semisimple groups over function fields, Ann. of Math. (2) 105 (1977), 553–572.CrossRefGoogle Scholar
[Pra82]Prasad, G., Elementary proof of a theorem of Bruhat–Tits–Rousseau and of a theorem of Tits, Bull. Soc. Math. France 110 (1982), 197–202.CrossRefGoogle Scholar
[Rus70]Russell, P., Forms of the affine line and its additive group, Pacific J. Math. 32 (1970), 527–539.CrossRefGoogle Scholar
[SS09]Sancho de Salas, C. and Sancho de Salas, F., Principal bundles, quasi-abelian varieties, and structure of algebraic groups, J. Algebra 322 (2009), 2751–2772.CrossRefGoogle Scholar
[Ser88]Serre, J.-P., Algebraic groups and class fields (Springer, New York, 1988).CrossRefGoogle Scholar
[Ser97]Serre, J.-P., Galois cohomology (Springer, New York, 1997).CrossRefGoogle Scholar
[Wei82]Weil, A., Adeles and algebraic groups (Birkhäuser, Boston, 1982).CrossRefGoogle Scholar
[Wil95]Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551.CrossRefGoogle Scholar