2024/12/07 by Pertile, Jordan, Starichkova, Valeriia V.
#11H31 #11R04 #11R42 #11Y35 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2412.05568
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 ≥ 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 OK. 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 ζK, a (potential) lower bound for ζK 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 ζK is that it can be computationally checked for number fields of not too large degrees.