Deformations of Galois Representations and the Theorems of Sato-Tate, Lang-Trotter and others

TL;DR

This paper constructs infinitely ramified Galois representations with atypical distribution properties, challenging classical conjectures like Sato-Tate and Lang-Trotter, and extends Ramakrishna's results on large image deformations.

Contribution

It introduces new methods to construct infinitely ramified Galois representations with unusual distribution and large image, expanding understanding of Galois deformation theory.

Findings

Constructed infinitely ramified Galois representations with non-standard distributions.

Deformed residual Galois representations to achieve large image and infinite ramification.

Challenged classical distribution conjectures for Galois representations.

Abstract

We construct infinitely ramified Galois representations $\rho$ such that the $a_l (\rho)$'s have distributions in contrast to the statements of Sato-Tate, Lang-Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalising a result of Ramakrishna.