# Constructing Certain Special Analytic Galois Extensions

## TL;DR

This paper constructs special $p$-adic analytic Galois extensions of cyclotomic fields with controlled ramification, expanding known cases through lifting reducible Galois representations for many primes.

## Contribution

It introduces a new method to construct analytic Galois extensions for numerous primes using lifting techniques of reducible Galois representations.

## Key findings

- Constructed extensions are unramified at $p$ and tamely ramified at finitely many primes.
- Extensions are isomorphic to finite index subgroups of $ ext{SL}_2(bZ_p)$ containing principal congruence subgroups.
- Method applies to many primes beyond previously known cases.

## Abstract

For every prime $p\geq 5$ for which a certain condition on the class group $\text{Cl}(\mathbb{Q}(\mu_p))$ is satisfied, we construct a $p$-adic analytic Galois extension of the infinite cyclotomic extension $\mathbb{Q}(\mu_{p^{\infty}})$ with some special ramification properties. In greater detail, this extension is unramified at primes above $p$ and tamely ramified above finitely many rational primes and is isomorphic to a finite index subgroup of $\text{SL}_2(\mathbb{Z}_p)$ which contains the principal congruence subgroup. For the primes $107,139,271$ and $379$ such extensions were first constructed by Ohtani and Blondeau. The strategy for producing these special extensions at an abundant number of primes is through lifting two-dimensional reducible Galois representations which are diagonal when restricted to $p$.