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Weil's conjecture on Tamagawa numbers

November 23, 2023

In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number of a simply connected simple algebraic group defined over a number field is 1. In this case, simply connected means "not having a proper algebraic covering" in the algebraic group theory sense, which is not always the topologists' meaning.

History edit

Weil (1959) calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. The first observation does not hold for all groups: Ono (1963) found examples where the Tamagawa numbers are not integers. The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.

Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups. K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. Kottwitz (1988) proved it for all groups satisfying the Hasse principle, which at the time was known for all groups without E8 factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E8 case (see strong approximation in algebraic groups), thus completing the proof of Weil's conjecture. In 2011, Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over function fields over finite fields.[1]

Applications edit

Ono (1965) used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups.

For spin groups, the conjecture implies the known Smith–Minkowski–Siegel mass formula.[1]

See also edit

References edit

  1. ^ a b Lurie 2014.
  • "Tamagawa number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Chernousov, V. I. (1989), "The Hasse principle for groups of type E8", Soviet Math. Dokl., 39: 592–596, MR 1014762
  • Kottwitz, Robert E. (1988), "Tamagawa numbers", Ann. of Math., 2, Annals of Mathematics, 127 (3): 629–646, doi:10.2307/2007007, JSTOR 2007007, MR 0942522.
  • Lai, K. F. (1980), "Tamagawa number of reductive algebraic groups", Compositio Mathematica, 41 (2): 153–188, MR 0581580
  • Langlands, R. P. (1966), "The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups", Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., pp. 143–148, MR 0213362
  • Ono, Takashi (1963), "On the Tamagawa number of algebraic tori", Annals of Mathematics, Second Series, 78 (1): 47–73, doi:10.2307/1970502, ISSN 0003-486X, JSTOR 1970502, MR 0156851
  • Ono, Takashi (1965), "On the relative theory of Tamagawa numbers", Annals of Mathematics, Second Series, 82 (1): 88–111, doi:10.2307/1970563, ISSN 0003-486X, JSTOR 1970563, MR 0177991
  • Tamagawa, Tsuneo (1966), "Adèles", Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., vol. IX, Providence, R.I.: American Mathematical Society, pp. 113–121, MR 0212025
  • Voskresenskii, V. E. (1991), Algebraic Groups and their Birational Invariants, AMS translation
  • Weil, André (1959), Exp. No. 186, Adèles et groupes algébriques, Séminaire Bourbaki, vol. 5, pp. 249–257
  • Weil, André (1982) [1961], Adeles and algebraic groups, Progress in Mathematics, vol. 23, Boston, MA: Birkhäuser Boston, ISBN 978-3-7643-3092-7, MR 0670072
  • Lurie, Jacob (2014), Tamagawa Numbers via Nonabelian Poincaré Duality

Further reading edit

  • Aravind Asok, Brent Doran and Frances Kirwan, "Yang-Mills theory and Tamagawa Numbers: the fascination of unexpected links in mathematics", February 22, 2013
  • J. Lurie, The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality posted June 8, 2012.

November 23, 2023
weil, conjecture, tamagawa, numbers, mathematics, weil, conjecture, tamagawa, numbers, statement, that, tamagawa, number, displaystyle, simply, connected, simple, algebraic, group, defined, over, number, field, this, case, simply, connected, means, having, pro. In mathematics the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number t G displaystyle tau G of a simply connected simple algebraic group defined over a number field is 1 In this case simply connected means not having a proper algebraic covering in the algebraic group theory sense which is not always the topologists meaning Contents 1 History 2 Applications 3 See also 4 References 5 Further readingHistory editWeil 1959 calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected The first observation does not hold for all groups Ono 1963 found examples where the Tamagawa numbers are not integers The second observation that the Tamagawa numbers of simply connected semisimple groups seem to be 1 became known as the Weil conjecture Robert Langlands 1966 introduced harmonic analysis methods to show it for Chevalley groups K F Lai 1980 extended the class of known cases to quasisplit reductive groups Kottwitz 1988 proved it for all groups satisfying the Hasse principle which at the time was known for all groups without E8 factors V I Chernousov 1989 removed this restriction by proving the Hasse principle for the resistant E8 case see strong approximation in algebraic groups thus completing the proof of Weil s conjecture In 2011 Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over function fields over finite fields 1 Applications editOno 1965 used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups For spin groups the conjecture implies the known Smith Minkowski Siegel mass formula 1 See also editTamagawa numberReferences edit a b Lurie 2014 Tamagawa number Encyclopedia of Mathematics EMS Press 2001 1994 Chernousov V I 1989 The Hasse principle for groups of type E8 Soviet Math Dokl 39 592 596 MR 1014762 Kottwitz Robert E 1988 Tamagawa numbers Ann of Math 2 Annals of Mathematics 127 3 629 646 doi 10 2307 2007007 JSTOR 2007007 MR 0942522 Lai K F 1980 Tamagawa number of reductive algebraic groups Compositio Mathematica 41 2 153 188 MR 0581580 Langlands R P 1966 The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups Algebraic Groups and Discontinuous Subgroups Proc Sympos Pure Math Providence R I Amer Math Soc pp 143 148 MR 0213362 Ono Takashi 1963 On the Tamagawa number of algebraic tori Annals of Mathematics Second Series 78 1 47 73 doi 10 2307 1970502 ISSN 0003 486X JSTOR 1970502 MR 0156851 Ono Takashi 1965 On the relative theory of Tamagawa numbers Annals of Mathematics Second Series 82 1 88 111 doi 10 2307 1970563 ISSN 0003 486X JSTOR 1970563 MR 0177991 Tamagawa Tsuneo 1966 Adeles Algebraic Groups and Discontinuous Subgroups Proc Sympos Pure Math vol IX Providence R I American Mathematical Society pp 113 121 MR 0212025 Voskresenskii V E 1991 Algebraic Groups and their Birational Invariants AMS translation Weil Andre 1959 Exp No 186 Adeles et groupes algebriques Seminaire Bourbaki vol 5 pp 249 257 Weil Andre 1982 1961 Adeles and algebraic groups Progress in Mathematics vol 23 Boston MA Birkhauser Boston ISBN 978 3 7643 3092 7 MR 0670072 Lurie Jacob 2014 Tamagawa Numbers via Nonabelian Poincare DualityFurther reading editAravind Asok Brent Doran and Frances Kirwan Yang Mills theory and Tamagawa Numbers the fascination of unexpected links in mathematics February 22 2013 J Lurie The Siegel Mass Formula Tamagawa Numbers and Nonabelian Poincare Duality posted June 8 2012 Retrieved from https en wikipedia org w index php title Weil 27s conjecture on Tamagawa numbers amp oldid 1134994116, wikipedia, wiki, book, books, library,

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