The size function for imaginary cyclic sextic fields

TL;DR

This paper proves that the size function $h^0$ reaches its maximum at the trivial class for totally imaginary cyclic sextic fields, extending previous results to this new class of number fields.

Contribution

It establishes the conjecture for the size function $h^0$ in totally imaginary cyclic sextic fields, a previously unverified class with rank two unit groups.

Findings

The conjecture holds for totally imaginary cyclic sextic fields.

The size function $h^0$ attains its maximum at the trivial class.

Extension of the conjecture to new classes of number fields.

Abstract

In this paper, we investigate the size function $h^0$ for number fields. This size function is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors. This conjecture was proved for all number fields with unit group of rank 0 and 1, and also for cyclic cubic fields which have unit group of rank two. We prove the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two.