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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)

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    $\begingroup$ You wrote $\text{GL}_2(\mathbb{Z}/p)$ in your first display. Did you mean mod $\ell$? The letter $p$ is not mentioned prior to this display. This makes a big difference. $\endgroup$ Feb 12 at 21:40
  • $\begingroup$ I edited that back to Z/ell, thank you! $\endgroup$
    – Uzu Lim
    Feb 12 at 21:46

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The slogan "number theorists aim to understand $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$" is one that gets used a lot, but it's perhaps a tiny little bit misleading.

Understanding the structure of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ as an abstract group, or maybe as a topological group, is of course a highly interesting problem; but the general verdict is that it's a huge chaotic jungle. E.g. the Inverse Galois problem is the conjecture that every finite group occurs as the quotient of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ by an open subgroup, generally in infinitely many ways.

Studying "restricted ramification" Galois groups -- quotients of $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ parametrizing extensions in which only a given finite set of primes is allowed to ramify -- cuts the jungle down to a managable size. These groups have tolerably explicit presentations in terms of (topological) generators and relations, as do the local groups $\operatorname{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$. But these abstract algebraic questions are somewhat disjoint from the mainstream of Galois-representation theory.

A more accurate, although much wordier, description of the main preoccupations of Galois representation theory might go as follows.

  • Galois representations "arise naturally" from arithmetically interesting objects (elliptic curves, modular forms, algebraic varieties).

  • Studying the Galois representations associated to these objects often provides a very powerful way of solving problems about their arithmetic, although those questions may initially seem to have nothing to do with Galois representations. Kolyvagin's proof of the Birch--Swinnerton-Dyer conjecture in analytic rank $\le 1$ is a classic example of this.

  • Studying the Galois representations gives a uniform way of understanding the arithmetic of many different classes of objects at the same time.

  • Conversely, all Galois representations which are "nice enough" -- expressed in terms of local conditions -- seem to arise from algebraic varieties (the Fontaine--Mazur conjecture).

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  • $\begingroup$ Your "wordier" summary is very helpful :) Thank you. I wrote a comment here before thanking you, but I think someone deleted it because I was simply "thanking" and not adding to the discussion, or whatever. Which is annoying, but fine, heh $\endgroup$
    – Uzu Lim
    Feb 13 at 22:45
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$\newcommand{\Q}{\mathbb{Q}} \DeclareMathOperator{\Gal}{Gal}$ Here is an example of theorem that may be in the style that you are looking for, in the sense that the statement does not involve Galois representations but its proof does. On the other hand, I think it illustrates well David Loeffler's point that number theorists are often interested in the Galois group not as an abstract topological group, but together with the ramification information coming from the completions of $\Q$.

Theorem (Chenevier - Clozel): Let $S$ be a finite set of primes, let $p\in S$ and let $\Q_S$ be the maximal algebraic extension of $\Q$ unramified outside of $S$. Assume $|S|\ge 2$. Then the natural map $$ \Gal(\bar{\Q}_p/\Q_p) \to \Gal(\Q_S/\Q) $$ is injective.

Note that the corresponding statement with $\Q_S$ replaced by $\bar\Q$ is easy and does not require any Galois representations at all!

Reference: G. Chenevier and L. Clozel, Corps de nombres peu ramifiés et formes automorphes autoduales, J. Amer. Math. Soc., 2009.

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  • $\begingroup$ So perhaps when people say "we want to understand the absolute Gal group", they also mean to understand the action on other arithmetic objects. Which happen to be vector spaces anyway. Thank you! $\endgroup$
    – Uzu Lim
    Feb 13 at 16:44
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    $\begingroup$ This is an excellent example. However, as far as I'm aware, it's also quite an unusual one: I'd go as far as to say that it's the only example I know of where automorphic forms and other heavyweight machinery from the Langlands program has been used to prove a result about Galois groups which is (almost) purely group-theoretic. $\endgroup$ Feb 13 at 21:38
  • $\begingroup$ @UzuLim Was that meant to be a comment on Aurel's answer, or on mine? $\endgroup$ Feb 13 at 21:38
  • $\begingroup$ @DavidLoeffler I agree it is unusual. There are also a few cases of the inverse Galois problem too, but the relation to Galois representations is more obvious. And Gross's problem on the unsolvability of $Gal(\mathbb{Q}_S/\mathbb{Q})$. But that's about all the examples I know. $\endgroup$
    – Aurel
    Feb 13 at 22:02

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