# Galois groups with prescribed ramification and non-abelian class field theory

## Brian Hwang (Cal Tech)

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 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 required to answer arithmetic questions, there is currently no known way to systematically construct such extensions in full generality.
However, there have been some programs that are gaining ground on this front. One method is to construct such Galois representations via the Langlands correspondence. We will explain this method, show how some recent advances give us additional control over the number fields constructed, and indicate how this brings us closer to our goal. As an application, we will show how one can use this knowledge to study the arithmetic of curves over number fields.

Tuesday, October 20th, at 4:00pm in HBH 227

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