The size function for imaginary cyclic sextic fields

The size function $h^0$ for a number field 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 if that field is Galois over $\mathbb{Q}$ or over an imaginary quadratic field. This conjecture was proved for all number fields with the unit group of rank $0$ and $1$, and also for cyclic cubic fields which have unit group of rank two. In this talk, we will discuss the main idea to prove that the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two. This is joint work with Peng Tian and Amy Feaver.