Increasing the Size of Tame Shafarevich Groups

TL;DR

This paper investigates how the second Shafarevich group can be enlarged by adding primes to the set S in number fields, extending previous results to more general modules.

Contribution

It generalizes prior work by showing the increase of a2 groups for broader modules and tame sets, using Liu's uB to demonstrate maximal growth.

Findings

a2 groups can attain maximal dimension with carefully chosen primes

Existence of infinitely many tame prime sets enlarging a2 groups

Extension of previous results to general a2 modules

Abstract

Let $K$ be a number field with a finite set $S$ of primes. We study the cohomology of $\mathbb{F}_p[G_{K,S}]$-modules $A$, in particular the Shafarevich groups $\Sha^i_S(K,A)$ for $i=1,2$ for tame sets $S$, i.e., for sets $S$ that contain no primes above $p$. When $S$ contains all primes above $p$ (the ``wild'' setting), it is a consequence of global Poitou-Tate duality that $\Sha^1_S(K,A')^\vee \simeq \Sha^2_S(K,A) \stackrel{\simeq}{\hookrightarrow} \RusB_S(K,A) $ is non-increasing as $S$ increases. The same applies when $G_{K,S}$ is replaced by its maximal pro-$p$ quotient $G_{K,S}(p)$. In [4] it was shown that for $S$ tame and $A=\mathbb{F}_p$ with trivial action, the group $\Sha^2_S(K, A)$ can increase as $S$ increases to $S\cup X$, and even attain its maximal dimension, $\dim_{\mathbb{F}_p} \RusB_S(K,\mathbb{F}_p)$, for carefully chosen $X$. We strengthen this to general $\mathbb{F}_p[G_{K,S}]$-modules $A$ where $S$ is tame. We use Liu's definition [7] of $\RusB_S(K,A)$ to show that $\Sha^2_S(K,A) \hookrightarrow \RusB_S(K,A)$ and that there exist infinitely many tame sets of primes $X$ of $K$ such that $\Sha^2_{S\cup X}(K,A) \stackrel{\simeq}{\hookrightarrow} \RusB_{S \cup X}(K,A) \stackrel{\simeq}{\twoheadleftarrow} \RusB_S(K,A) \hookleftarrow \Sha^2_S(K,A)$.