The (α, β)-ramification invariants of a number field

2019/06/10 by Guillermo Mantilla-Soler, Mantilla-Soler, Guillermo
Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.1906.04254

Abstract

Let L be a number field. For a given prime p we define integers α_p^L and β_p^L with some interesting arithmetic properties. For instance, β_p^L is equal to 1 whenever p does not ramify in L and α_p^L is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α_p^L is not zero for all p then such residues determine the genus of the integral trace.

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