On unit signatures and narrow class groups of odd degree abelian number fields. (English) Zbl 1518.11078
In this paper, the main object of research are algebraic number fields \(K\) with abelian Galois group \(G\) and odd degree over \(\mathbb Q\). The authors work out the connections between the \(2\)-Selmer group of \(K\), the unit signature rank of \(K\), and the \(2\)-ranks of the class group, the narrow class group and the ray class group of conductor \(4\) of \(K\). Each of these groups is a \(\mathbb Z [G]\)-module \(M\), and to analyze its \(2\)-rank one studies the \(\mathbb F_2 [G]\)-module \(M/M^2\). Each irreducible component of the latter is isomorphic to an extension field \(\mathbb F_2 (\chi) \subset \overline {\mathbb F}_2\), where \(\chi\) is an \(\overline {\mathbb F}_2\)-character of \(G\) and \(\overline {\mathbb F}_2\) a fixed algebraic closure of \(\mathbb F_2\).
Let \(S(K)\) denote the image of the \(2\)-Selmer group of \(K\) under the \(2\)-Selmer signature map, as introduced by D. S. Dummit and the third author [Proc. Lond. Math. Soc. (3) 117, No. 4, 682–726 (2018; Zbl 1457.11152)]. The main result (Theorem 5.4.2) shows that there are exactly 6 possible structures for the irreducible components of the \(\mathbb F_2 [G]\)-module \(S(K)\). Using the above mentioned connections, this enables the authors to obtain results on the \(2\)-rank of the \(\chi\)-part of the unit signature group (Theorem 5.5.2) and in case that \(G\) is cyclic of prime degree, also on the \(2\)-rank of the unit signature group.
Inspired by these results, the authors put up conjectures in the sense of Cohen-Lenstra heuristics, as, e.g., Conjecture 1.1.3: “As \(K\) ranges over cyclic number fields of degree \(7\) with odd class number, the probability that the narrow class number is also odd is \(7/9\).” To provide evidence for their conjectures, the authors present computations with all cyclic cubic (\(X=10^7\)) and cyclic septic (\(X \approx 244861\)) fields, with conductor up to \(X\). In an appendix (with Noam Elkies) it is proved that there indeed exist infinitely many cyclic cubic number fields with signature rank \(1\).
Let \(S(K)\) denote the image of the \(2\)-Selmer group of \(K\) under the \(2\)-Selmer signature map, as introduced by D. S. Dummit and the third author [Proc. Lond. Math. Soc. (3) 117, No. 4, 682–726 (2018; Zbl 1457.11152)]. The main result (Theorem 5.4.2) shows that there are exactly 6 possible structures for the irreducible components of the \(\mathbb F_2 [G]\)-module \(S(K)\). Using the above mentioned connections, this enables the authors to obtain results on the \(2\)-rank of the \(\chi\)-part of the unit signature group (Theorem 5.5.2) and in case that \(G\) is cyclic of prime degree, also on the \(2\)-rank of the unit signature group.
Inspired by these results, the authors put up conjectures in the sense of Cohen-Lenstra heuristics, as, e.g., Conjecture 1.1.3: “As \(K\) ranges over cyclic number fields of degree \(7\) with odd class number, the probability that the narrow class number is also odd is \(7/9\).” To provide evidence for their conjectures, the authors present computations with all cyclic cubic (\(X=10^7\)) and cyclic septic (\(X \approx 244861\)) fields, with conductor up to \(X\). In an appendix (with Noam Elkies) it is proved that there indeed exist infinitely many cyclic cubic number fields with signature rank \(1\).
Reviewer: Günter Lettl (Graz)
MSC:
| 11R27 | Units and factorization |
| 11R29 | Class numbers, class groups, discriminants |
| 11Y40 | Algebraic number theory computations |
| 11R80 | Totally real fields |
| 11R45 | Density theorems |
Keywords:
\(2\)-Selmer group; signature rank; ray class group; Galois module; reflection theorem; Cohen-Lenstra heuristicsCitations:
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