Crible et 3-rang des corps quadratiques
Annales de l'Institut Fourier, Tome 46 (1996) no. 4, p. 909-949
Considérons le cardinal h 3 * (Δ) de l’ensemble des racines cubiques de l’unité dans le groupe des classes de (Δ), où Δ est un discriminant fondamental. Un résultat de Davenport et Heilbronn calcule la valeur moyenne de ces nombres quand Δ varie. On obtient ici géométriquement une borne explicite pour le reste, avec la possibilité supplémentaire de restreindre les Δ à des progressions arithmétiques. Des techniques de crible permettent alors d’évaluer la 3-partie des (±P k ), où P k est pseudo-premier d’ordre k. On contrôle ainsi simultanément le 2-rang et le 3-rang du groupe des classes Cl ((Δ)). L’auteur donne en particulier une borne pour le 3-rang en moyenne des (±p), où p est premier.
Call h 3 * (Δ) the number of cube roots of unity in the class group of (Δ), where Δ is a fundamental discriminant. Davenport and Heilbronn computed the mean value of these numbers when Δ tends to ±. The author gives a general geometric argument yielding an explicit bound for the error term, with the additional possibility of restricting Δ to arithmetic progressions. Sieve techniques then produce results about the 3-parts of the groups Cl ((Δ)), where P k is an almost-prime of order k. In this way, one controls simultaneously both the 2-rank and the 3-rank of the class group Cl ((Δ)). As a special case, the author gives a bound for the mean 3-rank of the (±p), where p is prime.
@article{AIF_1996__46_4_909_0,
     author = {Belabas, Karim},
     title = {Crible et 3-rang des corps quadratiques},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {4},
     year = {1996},
     pages = {909-949},
     doi = {10.5802/aif.1535},
     zbl = {0853.11088},
     mrnumber = {98b:11112},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1996__46_4_909_0}
}
Belabas, Karim. Crible et 3-rang des corps quadratiques. Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 909-949. doi : 10.5802/aif.1535. https://www.numdam.org/item/AIF_1996__46_4_909_0/

[1] R. Benedetti and J.-J. Risler, Real algebraic and semi-algebraic sets, Hermann, 1990. | MR 91j:14045 | Zbl 0694.14006

[2] D.A. Buell, Binary quadratic forms, Springer-Verlag, 1989. | Zbl 0698.10013

[3] H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, in Number Theory, Noordwijkerhout 1983, Lecture Notes in Math. n° 1068, Springer-Verlag, 1984. | Zbl 0558.12002

[4] B. Datskovsky and D.J. Wright, Density of discriminants of cubic extensions, J. reine. angew. Math., 386 (1988), 116-138. | MR 90b:11112 | Zbl 0632.12007

[5] H. Davenport, On a principle of Lipschitz, J. Lond. Math. Soc., 26 (1951), 179-183. | MR 13,323d | Zbl 0042.27504

[6] H. Davenport, On the class number of binary cubic forms (I), J. Lond. Math. Soc., 26 (1951), 183-192 (erratum, ibid. 27 (1951), p. 512). | Zbl 0044.27002

[7] H. Davenport, On the class number of binary cubic forms (II), J. Lond. Math. Soc., 26 (1951), 192-198. | MR 13,323f | Zbl 0044.27002

[8] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields (I), Bull. Lond. Math. Soc., 1 (1969), 345-348. | MR 40 #7223 | Zbl 0211.38602

[9] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields (II), Proc. Roy. Soc. Lond. A, 322 (1971), 405-420. | MR 58 #10816 | Zbl 0212.08101

[10] E. Fouvry, Sur le comportement en moyenne du rang des courbes y2 = x3 + k, in Séminaire de Théorie des Nombres Paris, 1990-1991, Birkhäuser, 1993, 61-83. | Zbl 0814.11034

[11] H. Halberstam and H.E. Richert, Sieve methods, Academic Press, 1974. | MR 54 #12689 | Zbl 0298.10026

[12] H. Hasse, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörper-theoretischer Grundlage, Math. Zeitschrift, 31 (1930), 565-582. | JFM 56.0167.02

[13] H. Iwaniec, A new form of the error term in the linear sieve, Acta. Arith., 37 (1980), 307-320. | MR 82d:10069 | Zbl 0444.10038

[14] H. Iwaniec, Rosser's sieve, Acta. Arith., 36 (1980), 171-202. | MR 81m:10086 | Zbl 0435.10029

[15] N.M. Katz, Perversity and exponential sums, Adv. Stud. in Pure Math., 17 (1989), 210-259. | MR 92m:11080 | Zbl 0755.14008

[16] N.M. Katz and G. Laumon, Transformation de Fourier et majoration de sommes exponentielles, Pub. Math. IHES, 62 (1985), 361-418. | Numdam | MR 87i:14017 | Zbl 0603.14015

[17] G.-B. Mathews, On the reduction and classification of binary cubic which have a negative discriminant, Proc. London Math. Soc., 10 (1912), 128-138. | JFM 42.0243.04

[18] J. Quer, Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12, C. R. Acad. Sciences, série I Math., 305 (1987), 215-218. | MR 88j:11074 | Zbl 0622.14025

[19] M. Sato and T. Shintani, On zeta functions associated with prehomogenous vector spaces, Ann. of Math., 100 (1974), 131-170. | MR 49 #8969 | Zbl 0309.10014

[20] T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan, 24 (1972), 132-188. | MR 44 #6619 | Zbl 0227.10031

[21] T. Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, Sec. Ia, 22 (1975), 25-66. | MR 52 #5590 | Zbl 0313.10041

[22] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Pub. Inst. Élie Cartan, 1990. | Zbl 0788.11001

[23] H. Weyl, On the volume of tubes, Amer. J. of Math., 61 (1939), 461-472. | JFM 65.0796.01 | Zbl 0021.35503