Invariants for \(A_{4}\) fields and the Cohen-Lenstra heuristics (Q2876603)

The Cohen-Lenstra heuristics fail in some cases, for instance if there is a contribution from genus theory or in the case where the ground field contains \(p\)th roots of unity. In this paper the author discusses a deviation from the Cohen-Lenstra heuristics when roots of unity are present. In particular, he proposes an explanation for the discrepancy between the observed number of cyclic cubic fields whose 2-class group is \(C_{2}\times C_{2}\) and the number predicted by the Cohen-Lenstra heuristics, in term of an invariant, for an \(A_{4}\) field lying in a quotient of the Schur multiplier group. The author also shows that, in some cases, the definition of the invariant can be greatly simplified and he provides a computation of the invariant when the cubic field is ramified at exactly one prime, up to \(10^{8}\).