On the 3-rank of quadratic fields with prime or almost prime discriminant
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 \textit{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.).
scientific article; zbMATH DE number 1425219
scientific article