Binary Hermitian forms over a cyclotomic field

TL;DR

This paper computes the Voronoi polyhedron for binary Hermitian forms over a cyclotomic field and classifies perfect forms, aiding in understanding the cohomology of related arithmetic groups.

Contribution

It provides the first explicit computation of the Voronoi polyhedron and classification of perfect forms over a cyclotomic field, linking geometric and algebraic structures.

Findings

Computed the Voronoi polyhedron for binary Hermitian forms over Q(z)

Classified GL_2(O)-conjugacy classes of perfect forms

Facilitated cohomology calculations of GL_2(O)

Abstract

Let z be a primitive fifth root of unity and let F be the cyclotomic field F=Q(z). Let O be the ring of integers. We compute the Voronoi polyhedron of binary Hermitian forms over F and classify GL_2(O)-conjugacy classes of perfect forms. The combinatorial data of this polyhedron can be used to compute the cohomology of the arithmetic group GL_2(O) and Hecke eigen forms.