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On the 3-rank of quadratic fields with prime or almost prime discriminant. (Sur le 3-rang des corps quadratiques de discriminant premier ou presque premier.) (French) Zbl 1061.11503
There is a presumably very deep conjecture that there are infinitely many primes $$p \equiv 1\bmod 4$$ for which the quadratic field $${\mathbb Q}(\sqrt p \,)$$ has class number $$1$$. The authors’ principal result gives a comparatively modest but unconditional approach to this conjecture, by showing that there are infinitely many such $$p$$ for which the ideal class group $$\text{Cl}(\sqrt p \,)$$ of $${\mathbb Q}(\sqrt p \,)$$ has no element of order $$3$$. Their result is quantitative in that it shows that a positive proportion of these primes not exceeding $$x$$ possesses the stated property.
Let $$\Delta$$ be a fundamental discriminant, so that it is squarefree and is either $$\equiv 1 \bmod 4$$ or is $$4\beta$$, where $$\beta \not\equiv 1 \bmod 4$$. Let $$\Delta^ -(X)$$ denote the number of negative $$\Delta$$ with $$|\Delta| \leq X$$, and define $$\Delta^ +(X)$$ similarly. Let $$h_p^ *(\Delta)$$ denote the number of $$p$$th roots of unity in $$\text{Cl}(\sqrt \Delta \,)$$, so that $$h_p^ *(\Delta)=p^ {r_p(\Delta)}$$, where $$r_p$$ is the $$p$$-rank of the title. Let $${\mathcal H}(\Delta)= {1\over2}\bigl( h_3^ *(\Delta)-1 \bigr)$$. The authors establish an asymptotic expression for $$\sum_{\Delta \equiv 0\bmod q}{\mathcal H}(\Delta)$$ when $$q \leq X^ {3/44}$$ and the sum is restricted to $$\Delta$$ in one of $$\Delta^ \pm(X)$$. The exponent $$3\over44$$ is better than that appearing in the earlier paper of K. Belabas [Ann. Inst. Fourier 46, No. 4, 909–949 (1996; Zbl 0853.11088)].
As in the earlier paper, this information can be used as the input for a sifting argument. The authors now infer, for example, that there are infinitely many fundamental discriminants for which the $$p$$-ranks satisfy $$r_ 2(\Delta) \leq 6$$ and $$r_ 3(\Delta) \geq 1$$. By means of a sieve with weights the constant 6 can be replaced by 3.
(This is the same review as for Math. Rev.).

##### MSC:
 11R29 Class numbers, class groups, discriminants 11N36 Applications of sieve methods
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##### References:
 [1] Karim Belabas, Crible et $$3$$-rang des corps quadratiques , Ann. Inst. Fourier (Grenoble) 46 (1996), no. 4, 909-949. · Zbl 0853.11088 [2] K. Belabas, On the mean $$3$$-rank of quadratic fields , Compositio Math., à paraître. · Zbl 0929.11046 [3] K. Belabas, Variations sur un thème de Davenport-Heilbronn , thèse de Doctorat d’État, Université Bordeaux 1, 1996. [4] Riccardo Benedetti and Jean-Jacques Risler, Real algebraic and semi-algebraic sets , Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. · Zbl 0694.14006 [5] H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields , Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33-62. · Zbl 0558.12002 [6] H. Davenport, On the class-number of binary cubic forms. I , J. London Math. Soc. 26 (1951), 183-192, Erratum, J. London Math. Soc. 27 (1951), 512. · Zbl 0044.27002 [7] H. Davenport, On the class-number of binary cubic forms. II , J. London Math. Soc. 26 (1951), 192-198. · Zbl 0044.27002 [8] H. Davenport, On a principle of Lipschitz , J. London Math. Soc. 26 (1951), 179-183. · Zbl 0042.27504 [9] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields , Bull. London Math. Soc. 1 (1969), 345-348. · Zbl 0211.38602 [10] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II , Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405-420. · Zbl 0212.08101 [11] B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree , Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. · Zbl 0133.30202 [12] H. Halberstam and H.-E. Richert, Sieve methods , Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974, London Math. Soc. Monographs (N.S.) 4. · Zbl 0298.10026 [13] C. S. Herz, “Construction of class fields” , Seminar on Complex Multiplication, Lecture Notes in Math., vol. 21, Springer-Verlag, Berlin, 1966, VII-1-VII-21. · Zbl 0147.03902 [14] Henryk Iwaniec, Rosser’s sieve , Acta Arith. 36 (1980), no. 2, 171-202. · Zbl 0435.10029 [15] Nicholas M. Katz and Gérard Laumon, Transformation de Fourier et majoration de sommes exponentielles , Inst. Hautes Études Sci. Publ. Math. (1985), no. 62, 361-418. · Zbl 0603.14015 [16] G. B. Mathews, On the reduction and classification of binary cubics which have a negative discriminant , Proc. London Math. Soc. 10 (1912), 128-138. · JFM 42.0243.04 [17] Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers , Springer-Verlag, Berlin, 1990, 2e éd. · Zbl 0717.11045 [18] Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres , Cours Spécialisés [Specialized Courses], vol. 1, Société Mathématique de France, Paris, 1995, 2e éd. · Zbl 0880.11001
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