# Images of Galois representations in mod $p$ Hecke algebras

## TL;DR

This paper investigates the images of Galois representations associated with mod p Hecke algebras attached to modular forms, providing criteria to determine their structure and applications to number field extensions.

## Contribution

It offers a method to explicitly determine the image of Galois representations in mod p Hecke algebras under specific conditions, extending understanding of their structure and implications.

## Key findings

- Determined the image of Galois representations under certain algebraic conditions.
- Connected the structure of these images to the existence of specific number field extensions.

## Abstract

Let $(\mathbb{T}_f,\mathfrak{m}_f)$ denote the mod $p$ local Hecke algebra attached to a normalised Hecke eigenform $f$, which is a commutative algebra over some finite field $\mathbb{F}_q$ of characteristic $p$ and with residue field $\mathbb{F}_q$. By a result of Carayol we know that, if the residual Galois representation $\overline{\rho}_f:G_\mathbb{Q}\rightarrow\mathrm{GL}_2(\mathbb{F}_q)$ is absolutely irreducible, then one can attach to this algebra a Galois representation $\rho_f:G_\mathbb{Q}\rightarrow\mathrm{GL}_2(\mathbb{T}_f)$ that is a lift of $\overline{\rho}_f$. We will show how one can determine the image of $\rho_f$ under the assumptions that $(i)$ the image of the residual representation contains $\mathrm{SL}_2(\mathbb{F}_q)$, $(ii)$ that $\mathfrak{m}_f^2=0$ and $(iii)$ that the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow us to deduce the existence of certain $p$-elementary abelian extensions of big non-solvable number fields.