发表机构
Swarthmore College; University of Bristol; Michigan State University; University of Pittsburgh（斯沃斯莫尔学院; 布里斯托大学; 密歇根州立大学; 匹兹堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用tame族中Eisenstein同余存在性的新结果，解释了一个特定metabelian数域中素数分裂行为的计算观察，延续了Ribet开创的范式。
AI 中文摘要
Ribet在1976年对Herbrand定理逆命题的证明开创了代数数论中的一个新范式，它表明数域的无分歧阿贝尔伽罗瓦扩张可以通过与Eisenstein同余（即与Eisenstein级数同余的尖点本征形式）相关的伽罗瓦表示来构造。近年来，在考察此类Eisenstein同余的数量方面取得了进一步进展，显示出当存在许多这样的同余时，可以获得关于相关扩张算术结构的额外且更精细的信息。沿着这一思路，我们利用关于tame族中Eisenstein同余存在性的新结果，来解释关于某个特定metabelian数域中素数分裂行为的一个计算观察。
英文摘要
Ribet's 1976 proof of the converse to Herbrand's theorem created a new paradigm in algebraic number theory by illustrating that unramified abelian Galois extensions of number fields can be constructed using Galois representations associated to Eisenstein congruences, i.e., cuspidal eigenforms that are congruent to Eisenstein series. In recent years, further progress has been made in investigating the quantity of such Eisenstein congruences, showing that when there are many such congruences, additional and finer information can be gleaned about the arithmetic structure of the related extensions. In this vein, we use new results on the existence of Eisenstein congruences in tame families to explain a computational observation about splitting behavior of primes in a certain metabelian number field.
Comments24 pages
