The size function for cyclic cubic fields

The size function for a number field is an analogue of the dimension of the Riemann–Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was proved for many number fields with unit groups of rank one. Our research confirms that the conjecture also holds for cyclic cubic fields, which have unit groups of rank two.

Invariants for A4 fields and the Cohen–Lenstra heuristics

This article discusses deviations from the Cohen–Lenstra heuristics when roots of unity are present. In particular, we propose an explanation for the discrepancy between the observed number of cyclic cubic fields whose 2-class group is C2 × C2 and the number predicted by the Cohen–Lenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute the invariant when the cubic field is ramified at exactly one prime, up to 108.

Cyclic Cubic Fields of Given Conductor and Given Index

The number of cyclic cubic fields with a given conductor and a given index is determined.