Exceptional Primes in Notions of Arithmetic Similarity
Shaver Phagan
TL;DR
The paper addresses when notions of arithmetic similarity between number fields, defined by equality of splitting statistics $s(p,F)$ for cofinitely many primes, have empty exceptional sets. It develops a unified, elementary group-theoretic framework, anchored by a counting formula for finite cyclic-group actions that links orbit data to stabilizers and ramification. Using this, it shows emptiness for several classical notions (arithmetic, Kronecker, and weak Kronecker equivalence) and constructs explicit non-empty exceptional-set examples (ultra-coarse equivalence and a ramification-based statistic). It also clarifies how these notions relate to Galois-group structures via decomposition and inertia, and discusses associated Dirichlet-series behavior. Overall, the work provides a principled, ramification-sensitive toolkit for understanding when arithmetic similarity collapses to a strong equality of splitting data and when genuine exceptions occur, with potential implications for related areas in algebra and beyond.
Abstract
A notion of arithmetic similarity between number fields is defined by requiring equality of some arithmetic statistics over all but finitely many rational primes. The exceptional set is empty in all previously studied cases, but existing methods for proving this are eclectic. We give a group theoretic explanation for those cases and show that some notions of arithmetic similarity admit a non-empty exceptional set. Furthermore, we prove a formula for actions of finite cyclic groups on finite sets, which augments a classical result of Gassmann and might also be of independent interest.