Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]