Locally cyclic extensions with Galois group $GL_2(p)$

TL;DR

This paper constructs Galois extensions of the rational numbers with Galois group $GL_2(p)$ where all decomposition groups are cyclic, using elliptic curve Galois representations, for all primes $p$.

Contribution

It provides the first explicit construction of such extensions with Galois group $GL_2(p)$ and cyclic decomposition groups for every prime $p$.

Findings

Constructed Galois extensions with group $GL_2(p)$ for all primes $p$.

All decomposition groups in these extensions are cyclic.

First realization of such extensions for the entire prime spectrum.

Abstract

Using Galois representations attached to elliptic curves, we construct Galois extensions of $\mathbb{Q}$ with group $GL_2(p)$ in which all decomposition groups are cyclic. This is the first such realization for all primes $p$.