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Zbl 1061.11503
Belabas, K.; Fouvry, E.
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)
[J] Duke Math. J. 98, No. 2, 217-268 (1999). ISSN 0012-7094

There is a presumably very deep conjecture that there are infinitely many primes $p \equiv 1\bmod 4$ for which the quadratic field ${\Bbb Q}(\sqrt p \,)$ 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 ${\Bbb Q}(\sqrt p \,)$ 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 $\Delta\sp -(X)$ denote the number of negative $\Delta$ with $|\Delta| \le X$, and define $\Delta\sp +(X)$ similarly. Let $h_p\sp *(\Delta)$ denote the number of $p$th roots of unity in $\text{Cl}(\sqrt \Delta \,)$, so that $h_p\sp *(\Delta)=p\sp {r_p(\Delta)}$, where $r_p$ is the $p$-rank of the title. Let ${\cal H}(\Delta)= {1\over2}\bigl( h_3\sp *(\Delta)-1 \bigr)$. The authors establish an asymptotic expression for $\sum_{\Delta \equiv 0\bmod q}{\cal H}(\Delta)$ when $q \le X\sp {3/44}$ and the sum is restricted to $\Delta$ in one of $\Delta\sp \pm(X)$. The exponent $3\over44$ is better than that appearing in the earlier paper of {\it 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 $r\sb 2(\Delta) \le 6$ and $r\sb 3(\Delta) \ge 1$. By means of a sieve with weights the constant 6 can be replaced by 3. (This is the same review as for Math. Rev.).
[George Greaves (Cardiff)]
MSC 2000:
*11R29 Class numbers, class groups, discriminants
11N36 Appl. of sieve methods

Citations: Zbl 0853.11088

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