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Sieve and 3-rank of quadratic fields. (Crible et 3-rang des corps quadratiques.) (French) Zbl 0853.11088
Summary: Call \(h^*_3(\Delta)\) the number of cube roots of unity in the class group of \(\mathbb{Q}(\sqrt\Delta)\), where \(\Delta\) is a fundamental discriminant. Davenport and Heilbronn computed the mean value of these numbers when \(\Delta\) tends to \(\pm\infty\). The author gives a general geometric argument yielding an explicit bound for the error term, with the additional possibility of restricting \(\Delta\) to arithmetic progressions. Sieve techniques then produce results about the 3-parts of the groups \(\text{Cl}(\mathbb{Q}(\sqrt{\Delta}))\), 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 \(\text{Cl}(\mathbb{Q}(\sqrt{\Delta}))\). As a special case, the author gives a bound for the mean 3-rank of the \(\mathbb{Q} (\sqrt{\pm p})\), where \(p\) is prime.

MSC:
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11N36 Applications of sieve methods
11P21 Lattice points in specified regions
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References:
[1] R. BENEDETTI and J.-J. RISLER, Real algebraic and semi-algebraic sets, Hermann, 1990. · 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. · Zbl 0632.12007
[5] H. DAVENPORT, On a principle of Lipschitz, J. Lond. Math. Soc., 26 (1951), 179-183. · 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. · 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. · 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. · 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. · 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. · Zbl 0444.10038
[14] H. IWANIEC, Rosser’s sieve, Acta. Arith., 36 (1980), 171-202. · Zbl 0435.10029
[15] N.M. KATZ, Perversity and exponential sums, Adv. Stud. in Pure Math., 17 (1989), 210-259. · 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. · 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. · Zbl 0622.14025
[19] M. SATO and T. SHINTANI, On zeta functions associated with prehomogenous vector spaces, Ann. of Math., 100 (1974), 131-170. · 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. · 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. · 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
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