The size function for cyclic cubic fields

TL;DR

This paper proves that the size function for cyclic cubic fields reaches its maximum at the trivial class, extending previous results from rank one to rank two unit groups.

Contribution

It confirms the conjecture about the size function's maximum for cyclic cubic fields with rank two unit groups, expanding the class of fields where the conjecture holds.

Findings

The size function attains its maximum at the trivial class for cyclic cubic fields.

The conjecture is validated for fields with unit groups of rank two.

Supports the broader conjecture across more complex number fields.

Abstract

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.