# Faltings Serre method on three dimensional selfdual representations

## TL;DR

This paper refines the Faltings-Serre method to prove isomorphisms of 3-dimensional selfdual Galois representations over non-rational fields, exemplified by a specific case involving elliptic curves.

## Contribution

It introduces a refined Faltings-Serre approach applicable to 3-dimensional Galois representations over fields other than , demonstrating its effectiveness with a concrete example.

## Key findings

- Proved the isomorphism of a specific 3D Galois representation to a twisted symmetric square of an elliptic curve.
- Developed a refinement of the Faltings-Serre method for non- fields.
- Utilized  Lie algebra and Burnside groups in the proof.

## Abstract

We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to $3$-dimensional Galois representations with the ground field not equal to $\mathbb{Q}$. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.