最大弹性二次场

Pub Date : 2024-09-23 DOI:10.1016/j.jnt.2024.08.003
Paul Pollack
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引用次数: 0

摘要

回想一下,对于每个非零非单元都因数化为不可还原单元的域 R,R 的弹性定义如下{sr:π1⋯πr=ρ1⋯ρs,所有 πi,ρj 都不可还原}。如果 K 的整数环是 UFD,且{1,32,2,52,3,...}∪{∞}中的每个元素在 K 中作为无穷多阶的弹性出现,我们就称一个二次域 K 为最大弹性域。假设广义黎曼假说成立,我们证明 K=Q(2) 具有普遍弹性,并为最大弹性二次域的猜想特征提供证据。
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Maximally elastic quadratic fields
Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined assup{sr:π1πr=ρ1ρs, with all πi,ρj irreducible}. We call a quadratic field K maximally elastic if the ring of integers of K is a UFD and each element of {1,32,2,52,3,}{} appears as an elasticity of infinitely many orders inside K. This corresponds to the orders in K exhibiting, to the extent possible for a quadratic field, maximal variation in terms of the failure of unique factorization. Assuming the Generalized Riemann Hypothesis, we prove that K=Q(2) is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields.
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