Steinitz classes and partial genera of unimodular lattices over imaginary-quadratic fields

TL;DR

This paper classifies unimodular lattices over imaginary-quadratic fields, investigates partial class numbers and masses within genera, and establishes criteria for when partial genera contain only one class.

Contribution

It determines all possible genera of unimodular lattices over imaginary-quadratic fields and proves a mass formula for partial masses, addressing class number questions.

Findings

Counterexample showing partial class numbers can differ for Steinitz classes

Mass formula for partial masses of genera

Characterization of single-class partial genera

Abstract

In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occuring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is "no" by giving a counter example, while the answer to the second question is "yes" by proving a mass formula for partial masses. Finally, we determine a list of all single-class partial genera and show that a partial genus consists of only one class if and only if the whole genus consists of only one class.