A Primer on Counting Solutions to Central Embedding Problems

Alex Smith recently verified the distribution of the $2^\infty$-part of the class group of imaginary quadratic fields predicted by Gerth, Cohen, and Lenstra. Despite the fact that
$2$ is believed to be the exceptional case of the Cohen-Lenstra heuristics, we can prove  comparatively very little about the other prime parts of the class group. We see a similar situation arising is various analogs of the Cohen-Lenstra heuristics where some exceptional cases appear to be more accessible by current methods. Examples include the $\ell^\infty$-part of the class group of degree $\ell$ cyclic fields studied by Klys, Pagano, and Koymans as well as the Cohen-Lenstra moment for specific nonabelian 2-groups over quadratic fields studied
by Alberts and Klys.

The common theme in each of these situations is the central embedding problem. In this talk we will give an overview of the underlying Galois theory that makes the exceptional cases both exceptional and more accessible, and begin laying down the foundation for a more general
approach to these cases. This talk will be followed up by Jack Klys putting this foundation into action to produce results in the nonabelian analog of the Cohen-Lenstra heuristics.