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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 $p\equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$ for which the quadratic field $ℚ\left(\sqrt{p}\phantom{\rule{0.166667em}{0ex}}\right)$ 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 $\text{Cl}\left(\sqrt{p}\phantom{\rule{0.166667em}{0ex}}\right)$ of $ℚ\left(\sqrt{p}\phantom{\rule{0.166667em}{0ex}}\right)$ 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 $\equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$ or is $4\beta$, where $\beta ¬\equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$. Let ${{\Delta }}^{-}\left(X\right)$ denote the number of negative ${\Delta }$ with $|{\Delta }|\le X$, and define ${{\Delta }}^{+}\left(X\right)$ similarly. Let ${h}_{p}^{*}\left({\Delta }\right)$ denote the number of $p$th roots of unity in $\text{Cl}\left(\sqrt{{\Delta }}\phantom{\rule{0.166667em}{0ex}}\right)$, so that ${h}_{p}^{*}\left({\Delta }\right)={p}^{{r}_{p}\left({\Delta }\right)}$, where ${r}_{p}$ is the $p$-rank of the title. Let $ℋ\left({\Delta }\right)=\frac{1}{2}\left({h}_{3}^{*}\left({\Delta }\right)-1\right)$. The authors establish an asymptotic expression for ${\sum }_{{\Delta }\equiv 0\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}q}ℋ\left({\Delta }\right)$ when $q\le {X}^{3/44}$ and the sum is restricted to ${\Delta }$ in one of ${{\Delta }}^{±}\left(X\right)$. The exponent $\frac{3}{44}$ 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 ${r}_{2}\left({\Delta }\right)\le 6$ and ${r}_{3}\left({\Delta }\right)\ge 1$. By means of a sieve with weights the constant 6 can be replaced by 3.

(This is the same review as for Math. Rev.).

##### MSC:
 11R29 Class numbers, class groups, discriminants 11N36 Applications of sieve methods