Explicit computations in Iwasawa theory

TL;DR

This paper introduces two algorithms for computing layers of the anticyclotomic ${f Z}_3$-extension of imaginary quadratic fields, utilizing complex multiplication and Kummer theory, with applications to class group structures.

Contribution

It presents novel algorithms for explicit computation in Iwasawa theory, specifically for anticyclotomic extensions of imaginary quadratic fields.

Findings

Algorithms successfully compute layers of the anticyclotomic ${f Z}_3$-extension

Derived new results on class group structures of nonmaximal orders

Demonstrated the effectiveness of complex multiplication and Kummer theory methods

Abstract

We give two algorithms to compute layers of the anticyclotomic ${\bf Z}_3$-extension of an imaginary quadratic field. The first is based on complex multiplication techniques for nonmaximal orders; the second is based on Kummer theory. As an illustration of our results, we use the mirroring principle to derive results on the structure of class groups of nonmaximal orders.