I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as one of the most important objects of study, in particular because Tannakian philosophy states that they allow us to recover $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. So my question is rather simple: what are some "basic" things (if any) we can do it we really have a strong understanding of the structure of this group?

The question is meant to be flexible, meaning you can assume as much additional information as you want to consider "understanding" the group to mean (i.e. understanding the decomposition groups at every prime, etc.). However, I'm looking for a collection concrete applications of this sort of thing, in the spirit of number theory. Anything from solving diophantine equations, to statements about the distribution of primes/representations of primes, to reciprocity laws. Things a high school student could understand even if the machinery is rather complicated, or something you could tell another mathematician very far removed from number theory about what motivates this research.

One thing that comes to mind immediately is resolving the inverse Galois problem, but this is something that doesn't seem to yield a concrete application immediately (unless someone has an example, which I'd be interested to see). I'm curious to see if there's any sort of conjectured techniques to solving classical number theory problems that might become easier if we knew more about this group. Of course some speculation is allowed, as the actual (currently unknown) structure of the group might determine which applications are admissible.


1 Answer 1


Of course, Galois theory intervenes as a basic tool in the study of some diophantine equation. But deeper aspects, in the form of Galois representations, were crucial for the proof of at least 4 fantastic theorems in the last century.

  • The Mordell-Weil theorem concerning rational points of elliptic curves over the field of rational numbers (Mordell, 1922) or an abelian variety over a number field (Weil, 1929). It says that this group of rational points is finitely generated.

  • Galois representations also play an important role (via the Tate conjecture) in Faltings's proof (1983) of Mordell's conjecture.

  • The proof of Fermat's Last Theorem wouldn't have been possible without a deep understanding of properties representations of the group $\mathop{\rm Gal}(\bar{\bf Q}/{\bf Q})$. In fact, the core of the proof by Wiles/Taylor-Wiles (1995) consists in establishing the link between representations associated with elliptic curves and representations associated with modular forms.

  • Finally, Mihailescu's proof (2002) of Catalan conjecture, although more elementary than the previous ones, makes great use of cyclotomic fields and their class groups (hence, by class field theory, looks closely at the abelian quotients $\mathop{\rm Gal}(\bar{\bf Q}/{\bf Q})$).


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