Kummer's Criterion for Totally Real Number Fields

Perfect Forms over Totally Real Number Fields

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 Voronöı 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 f...

On Shintani’s ray class invariant for totally real number fields

We introduce a ray class invariant X(C) for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula X(C) = X1(C) · · ·Xn(C) where each Xi(C) corresponds to a real place (Theorem 3.5). Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices ...

Gross–Stark units for totally real number fields

Enumeration of Totally Real Number Fields of Bounded Root Discriminant

We enumerate all totally real number fields F with root discriminant δF ≤ 14. There are 1229 such fields, each with degree [F : Q] ≤ 9. In this article, we consider the following problem. Problem 1. Given B ∈ R>0, enumerate the set NF (B) of totally real number fields F with root discriminant δF ≤ B, up to isomorphism. To solve Problem 1, for each n ∈ Z>0 we enumerate the set NF (n,B) = {F ∈ NF...

Euclidean minima of totally real number fields: Algorithmic determination

This article deals with the determination of the Euclidean minimum M(K) of a totally real number field K of degree n ≥ 2, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree 2 to 8 and small discriminants, most of which were previously unknown. Tables are given at the end of this paper.