Universal quadratic forms and Dedekind zeta functions

TL;DR

This work studies universal quadratic forms over totally real number fields through Dedekind zeta values and Siegel's formula to obtain an explicit upper bound on the rank of a universal form under the assumption that the codifferent is generated by a totally positive element. The analysis combines lattice-short-vector bounds with Siegel's formula, yielding a rank bound that depends on the degree $d$, discriminant $\Delta_K$, and auxiliary functions $G(\Delta_K)$ and $B(d)$, and leads to finiteness results for fixed $d$ and rank. To address the necessity of the principal codifferent assumption, the authors also investigate the generation of the positive part of ideals, proving that the maximal required number of generators $\kappa(K)$ is finite and connecting $\kappa(I)$ in real quadratic fields to continued fraction data of $\xi_D$, with explicit bounds and examples. Overall, the paper links analytic invariants from zeta functions with algebraic properties of ideals and lattices to constrain universal forms across totally real fields and to illuminate the structure of indecomposables in the positive cone.

Abstract

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadratic forms over a given number field $K$, under the assumption that the codifferent of $K$ is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• proof : Proof of Theorem $\ref{['maintheorem']}$
• proof : Proof of Theorem $\ref{['theoremcorollary']}$
• Theorem 2.4
• Proposition 3.1