On the Distribution of Class Groups of Abelian Extensions

Given a finite abelian group $\Gamma$, we study the distribution of the $p$-part of the class group $\operatorname{Cl}(K)$ as $K$ varies over Galois extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$ with Galois group isomorphic to $\Gamma$. We first construct a discrete valuation ring $e\mathbb{Z}_p[\Gamma]$ for each primitive idempotent $e$ of $\mathbb{Q}_p[\Gamma]$, such that 1) $e\mathbb{Z}_p[\Gamma]$ is a lattice of the irreducible $\mathbb{Q}_p[\Gamma]$-module $e\mathbb{Q}_p[\Gamma]$, and 2) $e\mathbb{Z}_p[\Gamma]$ is naturally a quotient of $\mathbb{Z}_p[\Gamma]$. For every $e$, we study the distribution of $e\operatorname{Cl}(K):=e\mathbb{Z}_p[\Gamma] \otimes_{\mathbb{Z}_p[\Gamma]} \operatorname{Cl}(K)[p^{\infty}]$, and prove that there is an ideal $I_e$ of $e\mathbb{Z}_p[\Gamma]$ such that $e\operatorname{Cl}(K) \otimes (e\mathbb{Z}_p[\Gamma]/I_e)$ is too large to have finite moments, while $I_e \cdot e\operatorname{Cl}(K)$ should be equidistributed with respect to a Cohen--Lenstra type of probability measure. We give conjectures for the probability and moment of the distribution of $I_e\cdot e\operatorname{Cl}(k)$, and prove a weighted version of the moment conjecture in the function field case. Our weighted-moment technique is designed to deal with the situation when the function field moment, obtained by counting points of Hurwitz spaces, is infinite; and we expect that this technique can also be applied to study other bad prime cases. Our conjecture agrees with the Cohen--Lenstra--Martinet conjecture when $p\nmid |\Gamma|$, and agrees with the Gerth conjecture when $\Gamma=\mathbb{Z}/p\mathbb{Z}$. We also study the kernel of $\operatorname{Cl}(K) \to \bigoplus_e e\operatorname{Cl}(K)$, and show that the average size of this kernel is infinite when $p^2\mid |\Gamma|$.