Increasing the Size of Tame Shafarevich Groups
Let $K$ be a number field with $S$ a finite set 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 $i=1,2$ and 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. A similar result holds when $G_{K,S}$ is replaced by its maximal pro-$p$ quotient $G_{K,S}(p)$. In [5] it was shown that for $S$ tame and $A=\mathbb{F}_p$ with trivial action, the group $\Sha^2_S(K, \mathbb{F}_p)$ can increase as $S$ increases to $S \cup X$, and even attain its maximal dimension, $\dim \RusB_S(K,\mathbb{F}_p)$, for carefully chosen $X$. In the first part of this paper, we use Liu's definition [8] of $\RusB_S(K,A)$ for a general $\mathbb{F}_p[G_{K, S}]$-module $A$ to show, assuming $\Sha^1_{all}(K,A')=0$, that $\Sha^2_S(K,A) \hookrightarrow \RusB_S(K,A)$. This happens, for example, when the action of $G_{K,S}$ on $A$ is through a finite group of order prime to $p$. Under this extra assumption, we then strengthen the results of [5] to show that for any odd prime $p$ and any $\mathbb{F}_p[G_{K, S}]$-module $A$ with $S$ tame, there exist infinitely many tame sets of primes $X$ of $K$ such that $\Sha^2_{S\cup X}(K,A) \stackrel{\simeq}{\hookrightarrow} \RusB_{S \cup X}(K,A) \stackrel{\simeq}{\twoheadleftarrow} \RusB_S(K,A) \hookleftarrow \Sha^2_S(K,A)$.