PhD Advisor: Jacob Tsimerman
We prove several results concerning class groups of number fields and function fields.
Firstly we compute all the moments of the $p$-torsion in the first step of a filtration of the class group defined by Gerth \cite{gerthprank} for cyclic number fields of degree $p$, unconditionally for $p=3$ and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the $p=3$ case this gives the distribution of the $3$-torsion
of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kl$\ddot{\text{u}}$ners in their proof of the distribution of the $4$-torsion in quadratic fields \cite{fk1}.
Secondly, we compute all the moments of a normalization of the function which counts unramified $H_{8}$-extensions of quadratic number fields, where $H_{8}$ is the quaternion group of order $8$, and show that the values of this function determine a constant distribution. Furthermore we propose
a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified $G$-extensions of quadratic fields for several other 2-groups $G$, which we conjecture will give finite moments which
determine a distribution. These are all cases in which the unnormalized average is known or conjectured to be infinite. Our method additionally can be used to determine the asymptotics of the unnormalized counting
function, which we also do for unramified $H_{8}$-extensions. This part of the thesis is joint work with Brandon Alberts.
Thirdly we present several new examples of reflection principles which apply to both class groups of number fields and picard groups of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer \cite{franz1} about equality of 2-rank in subfields of $A_{4}$, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group $G$ for the cases when $G$ is $S_{3}$, $S_{4}$, $A_{4}$, $D_{2l}$ and \mathbb{Z}/l\mathbb{Z}\rtimes\mathbb{Z}/r\mathbb{Z}$.
The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.