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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 p1mod4 for which the quadratic field (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 Cl(p) of (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 Δ be a fundamental discriminant, so that it is squarefree and is either 1mod4 or is 4β, where β¬1mod4. Let Δ - (X) denote the number of negative Δ with |Δ|X, and define Δ + (X) similarly. Let h p * (Δ) denote the number of pth roots of unity in Cl(Δ), so that h p * (Δ)=p r p (Δ) , where r p is the p-rank of the title. Let (Δ)=1 2h 3 * (Δ) - 1. The authors establish an asymptotic expression for Δ0modq (Δ) when qX 3/44 and the sum is restricted to Δ in one of Δ ± (X). The exponent 3 44 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 (Δ)6 and r 3 (Δ)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:
11R29Class numbers, class groups, discriminants
11N36Applications of sieve methods