Paramétrisation de structures algébriques et densité de discriminants
[Parametrization of algebraic structures and density of discriminants]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 935, pp. 267-299.

Gauss composition yields a group structure on the orbits of integer binary quadratic forms of discriminant D, modulo the natural SL 2 action. In essence, it is the class group of the quadratic order of discriminant D. Associated fundamental domains allow explicit computations and asymptotic evaluation of average orders. I shall present the higher composition laws discovered by M. Bhargava, their roots in the theory of regular prehomogeneous vector spaces, as well as the density results he obtains or conjectures, in particular concerning discriminants of algebraic number fields.

La composition de Gauss donne une structure de groupe aux orbites de formes quadratiques binaires entières de discriminant D, sous l’action de SL 2 par changement de variable, essentiellement le groupe des classes de l’ordre quadratique de discriminant D. Les domaines fondamentaux associés permettent calculs explicites et évaluation d’ordres moyens. Je présenterai les lois de composition supérieures découvertes par M. Bhargava à partir de la classification des espaces vectoriels préhomogènes réguliers, ainsi que les résultats de densité qu’il obtient ou conjecture, en particulier sur les discriminants de corps de nombres.

Classification: 11R04,  11R45,  11R29
Keywords: prehomogeneous vector space, density, discriminant, composition laws, number rings
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Belabas, Karim. Paramétrisation de structures algébriques et densité de discriminants, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 935, pp. 267-299. http://archive.numdam.org/item/SB_2003-2004__46__267_0/

[1] K. Belabas - “Crible et 3-rang des corps quadratiques”, Ann. Inst. Fourier (Grenoble) 46 (1996), p. 909-949. | Numdam | MR | Zbl

[2] M. Bhargava - “Higher composition laws”, Thèse, Princeton University, 2001. | MR | Zbl

[3] -, “Gauss composition and generalizations”, in Ants V, Sydney, Lecture Notes in Comput. Sci., no. 2369, Springer-Verlag, 2002, p. 1-9. | MR

[4] -, “The parametrization of algebraic structures”, Explicit Methods in Number Theory (Oberwolfach). Exposé, 2003.

[5] -, “Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations”, Ann. of Math. (2) 159 (2004), no. 1, p. 217-250. | MR | Zbl

[6] -, “Higher composition laws. II. On cubic analogues of Gauss composition”, Ann. of Math. (2) 159 (2004), no. 2, p. 865-886. | MR | Zbl

[7] -, “Higher composition laws. III. The parametrization of quartic rings”, Ann. of Math. (2) 159 (2004), no. 3, p. 1329-1360. | MR | Zbl

[8] D. A. Buell - Binary quadratic forms, Springer-Verlag, 1989. | MR | Zbl

[9] F. Chamizo & H. Iwaniec - “On the Gauss mean-value formula for class number”, Nagoya Math. J. 151 (1998), p. 199-208. | MR | Zbl

[10] H. Cohen - A course in computational algebraic number theory, 3e 'ed., Springer-Verlag, 1996. | MR | Zbl

[11] -, “Constructing and counting number fields”, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, p. 129-138. | MR | Zbl

[12] H. Cohen, F. Diaz Y Diaz & M. Olivier - “Enumerating quartic dihedral extensions of , Compositio Math. 133 (2002), no. 1, p. 65-93. | MR | Zbl

[13] -, “On the density of discriminants of cyclic extensions of prime degree”, J. reine angew. Math. 550 (2002), p. 169-209. | MR | Zbl

[14] H. Cohen & H. W. Lenstra, Jr. - “Heuristics on class groups of number fields”, in Number theory, Noordwijkerhout 1983, Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, p. 33-62. | MR | Zbl

[15] H. Cohen & J. Martinet - “Études heuristiques des groupes de classes des corps de nombres”, J. reine angew. Math. 404 (1990), p. 39-76. | MR | Zbl

[16] D. Cox - Primes of the form x 2 +ny 2 , Wiley-Interscience, 1989. | MR | Zbl

[17] B. Datskovsky & D. J. Wright - “The adelic zeta function associated to the space of binary cubic forms. II. Local theory”, J. reine angew. Math. 367 (1986), p. 27-75. | MR | Zbl

[18] -, “Density of discriminants of cubic extensions”, J. reine angew. Math. 386 (1988), p. 116-138. | MR | Zbl

[19] B. A. Datskovsky - “On Dirichlet series whose coefficients are class numbers of binary quadratic forms”, Nagoya Math. J. 142 (1996), p. 95-132. | MR | Zbl

[20] H. Davenport - “On a principle of Lipschitz”, J. London Math. Soc. 26 (1951), p. 179-183, corrigendum ibid 39 (1964), p. 580. | MR | Zbl

[21] H. Davenport & H. Heilbronn - “On the density of discriminants of cubic fields (ii)”, Proc. Roy. Soc. London Ser. A 322 (1971), p. 405-420. | MR | Zbl

[22] B. N. Delone & D. K. Faddeev - The theory of irrationalities of the third degree, Translations of Math. Monographs, vol. 10, American Mathematical Society, 1964. | MR | Zbl

[23] G. Eisenstein - “Untersuchungen über die cubischen Formen mit zwei Variabeln”, J. reine angew. Math. 27 (1844), p. 89-104. | Zbl

[24] J. Ellenberg & A. Venkatesh - “The number of extensions of a number field with fixed degree and bounded discriminant”, Ann. of Math., à paraître. | Zbl

[25] W. T. Gan, B. Gross & G. Savin - “Fourier coefficients of modular forms on G 2 , Duke Math. J. 115 (2002), no. 1, p. 105-169. | MR | Zbl

[26] D. Goldfeld & J. Hoffstein - “Eisenstein series of 1 2-integral weight and the mean value of real Dirichlet L-series”, Invent. Math. 80 (1985), no. 2, p. 185-208. | MR | Zbl

[27] J. W. Hoffman & J. Morales - “Arithmetic of binary cubic forms”, Enseign. Math. (2) 46 (2000), no. 1-2, p. 61-94. | MR | Zbl

[28] M. N. Huxley - “Integer points, exponential sums and the Riemann zeta function”, in Number theory for the millennium, II (Urbana, IL, 2000), AK Peters, Natick, MA, 2002, p. 275-290. | MR | Zbl

[29] J.-i. Igusa - An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2000. | MR | Zbl

[30] A. C. Kable & A. Yukie - “Prehomogeneous vector spaces and field extensions. II”, Invent. Math. 130 (1997), no. 2, p. 315-344. | MR | Zbl

[31] M. Kashiwara - B-functions and holonomic systems. Rationality of roots of B-functions”, Invent. Math. 38 (1976/77), no. 1, p. 33-53. | MR | Zbl

[32] T. Kimura - “The b-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces”, Nagoya Math. J. 85 (1982), p. 1-80. | MR | Zbl

[33] J. Klüners - “On the asymptotics of number fields with given Galois group”, 2003, Explicit Methods in Number Theory (Oberwolfach). Exposé.

[34] J. Klüners & G. Malle - “Counting nilpotent Galois extensions” | Zbl

[35] P. G. Lejeune Dirichlet - Vorlesungen über Zahlentheorie, Herausgegeben und mit Zusätzen versehen von R. Dedekind. Vierte, umgearbeitete und vermehrte Auflage, Chelsea Publishing Co., New York, 1968. | MR

[36] R. Lipschitz - “Über die asymptotischen Gesetze von gewissen Gattungen zahlentheoretischer Funktionen”, Monatsber. der Berl. Acad (1865), §174sqq.

[37] S. Mäki - On the density of abelian number fields, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, vol. 54, 1985. | MR | Zbl

[38] G. Malle - “On the distribution of Galois groups”, J. Number Theory 92 (2002), no. 2, p. 315-329. | MR | Zbl

[39] -, “On the distribution of Galois groups, II”, preprint, 2002. | MR

[40] D. P. Roberts - “Density of cubic field discriminants”, Math. Comp. 70 (2001), no. 236, p. 1699-1705. | MR | Zbl

[41] H. Rubenthaler - “Formes réelles des espaces préhomogènes irréductibles de type parabolique”, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 1, p. 1-38. | Numdam | MR | Zbl

[42] H. Saito - “On a classification of prehomogeneous vector spaces over local and global fields”, J. Algebra 187 (1997), no. 2, p. 510-536. | MR | Zbl

[43] -, “Convergence of the zeta functions of prehomogeneous vector spaces”, Nagoya Math. J. 170 (2003), p. 1-31. | MR | Zbl

[44] M. Sato - “Theory of prehomogeneous vector spaces (algebraic part)-the English translation of Sato's lecture from Shintani's note”, Nagoya Math. J. 120 (1990), p. 1-34, traduit du japonais par M. Muro. | MR | Zbl

[45] M. Sato & T. Kimura - “A classification of irreducible prehomogeneous vector spaces and their relative invariants”, Nagoya Math. J. 65 (1977), p. 1-155. | MR | Zbl

[46] M. Sato & T. Shintani - “On zeta functions associated with prehomogenous vector spaces”, Ann. of Math. 100 (1974), p. 131-170. | MR | Zbl

[47] J.-P. Serre - “Une “formule de masse” pour les extensions totalement ramifiées de degré donné d'un corps local”, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 22, p. A1031-A1036. | MR | Zbl

[48] D. Shanks - “On Gauss and composition. I, II”, in Number theory and applications (Banff, AB, 1988), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, p. 163-178, 179-204. | MR | Zbl

[49] T. Shintani - “On Dirichlet series whose coefficients are class numbers of integral binary cubic forms”, J. Math. Soc. Japan 24 (1972), p. 132-188. | MR | Zbl

[50] -, “On zeta-functions associated with the vector space of quadratic forms”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), p. 25-66. | MR | Zbl

[51] C. L. Siegel - “The average measure of quadratic forms with given determinant and signature”, Ann. of Math. (2) 45 (1944), p. 667-685. | MR | Zbl

[52] È. B. Vinberg - “The classification of nilpotent elements of graded Lie algebras”, Dokl. Akad. Nauk SSSR 225 (1975), no. 4, p. 745-748. | MR | Zbl

[53] S. Wong - “Automorphic forms on GL (2) and the rank of class groups”, J. reine angew. Math. 515 (1999), p. 125-153. | MR | Zbl

[54] D. J. Wright - “The adelic zeta function associated to the space of binary cubic forms. I. Global theory”, Math. Ann. 270 (1985), no. 4, p. 503-534. | MR | Zbl

[55] -, “Distribution of discriminants of abelian extensions”, Proc. London Math. Soc. (3) 58 (1989), no. 1, p. 17-50. | MR | Zbl

[56] D. J. Wright & A. Yukie - “Prehomogeneous vector spaces and field extensions”, Invent. Math. 110 (1992), no. 2, p. 283-314. | MR | Zbl

[57] A. Yukie - Shintani zeta functions, LMS Lect. Notes Series, vol. 183, Cambridge University Press, Cambridge, 1993. | MR | Zbl

[58] D. Zagier - “Cubic forms and cubic rings”, 2001, Explicit Methods in Number Theory (Oberwolfach). Exposé.