Reciprocity maps with restricted ramification

TL;DR

This paper compares two cohomological maps in number fields with restricted ramification, revealing new relationships between cup products, cyclotomic units, and Iwasawa modules, advancing understanding in algebraic number theory.

Contribution

It introduces and analyzes higher analogues of the S-reciprocity map, linking their cokernels to Iwasawa module filtrations, and explores their applications in p-ramified cohomology.

Findings

Relationship between cup products and cyclotomic p-units.

Higher reciprocity maps relate to Iwasawa module filtrations.

Cokernels of these maps connect to graded quotients in Iwasawa theory.

Abstract

We compare two maps that arise in study of the cohomology of number fields with ramification restricted to a finite set S of primes. One of these maps, which we call an S-reciprocity map, interpolates the values of cup products in S-ramified cohomology. In the case of p-ramified cohomology of the pth cyclotomic field for an odd prime p, we use this to exhibit an intriguing relationship between particular values of the cup product on cyclotomic p-units. We then consider higher analogues of the S-reciprocity map and relate their cokernels to the graded quotients in augmentation filtrations of Iwasawa modules.