Perfect forms over totally real number fields

TL;DR

This paper explores the classification of perfect quadratic forms over totally real number fields, providing methods to find initial forms and computing binary perfect forms over certain real quadratic fields.

Contribution

It introduces a method to find initial perfect forms over totally real fields and computes binary perfect forms over real quadratic fields with discriminant up to 66.

Findings

Method to find initial perfect forms for totally real fields

Explicit classification of binary perfect forms over Q(√d) for d ≤ 66

Finite classification of perfect forms up to equivalence

Abstract

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and later generalized by Koecher to arbitrary number fields. One knows that up to a natural "change of variables'' equivalence, there are only finitely many perfect forms, and given an initial perfect form one knows how to explicitly compute all perfect forms up to equivalence. In this paper we investigate perfect forms over totally real number fields. Our main result explains how to find an initial perfect form for any such field. We also compute the inequivalent binary perfect forms over real quadratic fields Q(\sqrt{d}) with d \leq 66.