On the Distribution of Class Groups of Abelian Extensions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915039668076544 |
|---|---|
| author | Liu, Yuan |
| author_facet | Liu, Yuan |
| contents | Given a finite abelian group $Γ$, 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 $Γ$. We first construct a discrete valuation ring $e\mathbb{Z}_p[Γ]$ for each primitive idempotent $e$ of $\mathbb{Q}_p[Γ]$, such that 1) $e\mathbb{Z}_p[Γ]$ is a lattice of the irreducible $\mathbb{Q}_p[Γ]$-module $e\mathbb{Q}_p[Γ]$, and 2) $e\mathbb{Z}_p[Γ]$ is naturally a quotient of $\mathbb{Z}_p[Γ]$. For every $e$, we study the distribution of $e\operatorname{Cl}(K):=e\mathbb{Z}_p[Γ] \otimes_{\mathbb{Z}_p[Γ]} \operatorname{Cl}(K)[p^{\infty}]$, and prove that there is an ideal $I_e$ of $e\mathbb{Z}_p[Γ]$ such that $e\operatorname{Cl}(K) \otimes (e\mathbb{Z}_p[Γ]/I_e)$ 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 $p\nmid |Γ|$, and agrees with the Gerth conjecture when $Γ=\mathbb{Z}/p\mathbb{Z}$. 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 $p^2\mid |Γ|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19318 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Distribution of Class Groups of Abelian Extensions Liu, Yuan Number Theory Given a finite abelian group $Γ$, 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 $Γ$. We first construct a discrete valuation ring $e\mathbb{Z}_p[Γ]$ for each primitive idempotent $e$ of $\mathbb{Q}_p[Γ]$, such that 1) $e\mathbb{Z}_p[Γ]$ is a lattice of the irreducible $\mathbb{Q}_p[Γ]$-module $e\mathbb{Q}_p[Γ]$, and 2) $e\mathbb{Z}_p[Γ]$ is naturally a quotient of $\mathbb{Z}_p[Γ]$. For every $e$, we study the distribution of $e\operatorname{Cl}(K):=e\mathbb{Z}_p[Γ] \otimes_{\mathbb{Z}_p[Γ]} \operatorname{Cl}(K)[p^{\infty}]$, and prove that there is an ideal $I_e$ of $e\mathbb{Z}_p[Γ]$ such that $e\operatorname{Cl}(K) \otimes (e\mathbb{Z}_p[Γ]/I_e)$ 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 $p\nmid |Γ|$, and agrees with the Gerth conjecture when $Γ=\mathbb{Z}/p\mathbb{Z}$. 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 $p^2\mid |Γ|$. |
| title | On the Distribution of Class Groups of Abelian Extensions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.19318 |