Extension of conjectures of Cohen and Lenstra. (English) Zbl 0613.12003
Let \(p\) be a prime number, and let \(K\) be a Galois extension of degree \(p\) of the field of rational numbers \({\mathbb Q}\). Let \(C\) be the ideal class group of \(K\). Let \(\sigma\) be a generator of \(\text{Gal}(K/ \mathbb Q)\), and let \(C^{1- \sigma}=\{a^{1-\sigma}: a\in C\}\). Suppose exactly \(t\) finite primes ramify in \(K/\mathbb Q\), where \(t\) is a positive integer. Then the genus group \(C/C^{1-\sigma}\) is an elementary abelian \(p\)-group with rank equal to \(t-1\) (or possibly \(t-2\) if \(K\) is real quadratic), and \(C^{1-\sigma}\) is called the principal genus. Let \(H=C^{1-\sigma}\), and let \(R\) be the \(p\)-rank of \(H/H^{1-\sigma}.\)
This paper states theorems which indicate how likely it is that \(R=0,1,2,... \). The formulas in these theorems are analogous to conjectured formulas of Cohen and Lenstra for the prime-to-\(p\) part of \(C\), thus suggesting that the Cohen-Lenstra conjectures [H. Cohen and H. W. Lenstra jun., Heuristics on class groups of number fields. Number theory, Proc. Journ. arith., Noodwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 33–62 (1984; Zbl 0558.12002)] might be extended to the principal genus in the ideal class groups of Galois extensions of \(\mathbb Q\) of prime degree \(p\).
This paper states theorems which indicate how likely it is that \(R=0,1,2,... \). The formulas in these theorems are analogous to conjectured formulas of Cohen and Lenstra for the prime-to-\(p\) part of \(C\), thus suggesting that the Cohen-Lenstra conjectures [H. Cohen and H. W. Lenstra jun., Heuristics on class groups of number fields. Number theory, Proc. Journ. arith., Noodwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 33–62 (1984; Zbl 0558.12002)] might be extended to the principal genus in the ideal class groups of Galois extensions of \(\mathbb Q\) of prime degree \(p\).
Reviewer: F. Gerth
MSC:
11R29 | Class numbers, class groups, discriminants |
11R11 | Quadratic extensions |
11R20 | Other abelian and metabelian extensions |