On Lenstra's criterion for norm-Euclideanity of number fields and properties of Dedekind zeta-functions

TL;DR

This work analyzes Lenstra's norm-Euclideanity criterion for number fields, establishing explicit bounds that render the criterion ineffective for large degrees under GRH by linking the minimal ideal-norm bound to a sphere-packing constant via the Rogers bound. It provides explicit upper and lower bounds for the Rogers constant $\sigma_n$, showing $\delta^*_2(n) \le \delta^*_1(n)$ for $n \ge 56$, and uses Poitou's discriminant bounds to derive a concrete threshold, $n \ge 62238$, beyond which Lenstra's criterion cannot hold under GRH. The authors also propose an alternative path: a lower bound on $\zeta_K(1+\beta)$ could replace GRH in obtaining similar contradictions, with a version of this idea supported by Zimmert-type arguments and explicit lemmas. The paper includes computational appendices with cyclotomic-field plots and discusses open questions about zeta-value bounds in families of number fields, suggesting practical avenues for testing the criteria in moderately large degrees.

Abstract

In 1977, Lenstra provided a criterion for norm-Euclideanity of number fields and noted that this criterion becomes ineffective for number fields of large enough degrees under the Generalised Riemann Hypothesis (GRH) for the Dedekind zeta-functions. In the first part of the paper, we provide an explicit version of this statement: the Lenstra criterion becomes ineffective under GRH for all number fields $K$ of degrees $n \geq 62238$. This follows from combining the criterion assumption with the explicit lower bound for the discriminant of $K$ under GRH, and the (trivial) upper bound for the minimal proper ideal norm in $\mathcal{O}_K$. Unconditionally, the lower bound for the discriminant is too weak to lead to such a contradiction. However, GRH can be replaced by another condition on the Dedekind zeta functions $ζ_K$, a (potential) lower bound for $ζ_K$ at a point on the right of $s = 1$. This condition combined with Zimmert's approach imply stronger upper bounds for the minimal proper ideal norm and again, contradicts to Lenstra's criterion for all $n$ large enough. The advantage of the new potential condition on $ζ_K$ is that it can be computationally checked for number fields of not too large degrees.

Figures (3)

• Figure 1: $\varepsilon = \frac{1}{4}$
• Figure 2: $\varepsilon = \frac{1}{2}$
• Figure 3: $\varepsilon = \frac{3}{4}$

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.1
• Theorem 1.3
• Theorem 2.1
• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof