Increasing the Size of Tame Shafarevich Groups

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 $\Sha^2_{S\cup X}(K,A) \stackrel{\simeq}{\hookrightarrow} \RusB_{S \cup X}(K,A) \stackrel{\simeq}{\twoheadleftarrow} \RusB_S(K,A) \hookleftarrow \Sha^2_S(K,A)$.