**Background.** A major theme of modern number theory is to study the absolute Galois group $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$. Galois representation theory attempts to understand $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$ via its representations. Here is one example that I just learned. Given an elliptic curve $E$ over $\mathbb{Q}$ and a prime number $\ell$, then there is a Galois representation given by:
$$ \rho_{E, \ell} : \text{Gal}(\overline{\mathbb Q} / \mathbb Q) \rightarrow \text{Aut}(E[\ell]) \cong \text{GL}_2(\mathbb Z/\ell) $$
where $E[\ell]$ is the group of $\ell$-torsion points of $E(\overline{\mathbb Q})$. Serre conjectured that there is a constant $\ell_0$ such that for every $\ell > \ell_0$ and non-CM elliptic curve $E$, the map $\rho_{E, \ell}$ is surjective. Now, all this makes me wonder as a "tourist" to number theory:

**Question 1.** What have mathematicians learned about $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$ itself through Galois representations? In other words, what are some major theorems from Galois representation theory and how did that help understand $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$?

**Question 2.** What are some major conjectures in Galois representation, and what does that make us believe about $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$ ?

I understand that there is a $p$-adic version of Galois representation, which I believe is a local program to apply the local-global principle to understand $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$. I'm happy to receive answers about $p$-adic Galois representation as well.

(I am an applied mathematician with a keen interest in number theory)