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A classical problem in Galois theory is a strong variant of

the Inverse Galois Problem: "What finite groups arise as the Galois

group of a finite Galois extension of the rational numbers, if you

impose the additional condition that the extension can only ramify at

finite set of primes?" This question is wide open in almost every

nonabelian case, and one reason is our lack of knowledge about how to

find number fields with prescribed ramification at fixed primes. While

such fields are often constructed to answer arithmetic questions,

there is currently no known way to systematically construct such

extensions in full generality.

However, there are some inspiring programs that are gaining ground on

this front. One method, initiated by Chenevier, is to construct such

number fields using Galois representations and their associated

automorphic representations via the Langlands correspondence. We will

explain the method, show how some recent advances in these subfields

allow us to gain some additional control over the number fields

constructed, and indicate how this brings us closer to our goal. As a

application, we will show how one can use this knowledge to study the

arithmetic of curves over number fields.