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 ${\text{Sha}}_S^i(K,A)$ 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 ${\text{Sha}}_S^1(K,A^{\mathrm{\prime }}{)}^{\vee }\simeq {\text{Sha}}_S^2(K,A){\text{RusB}}_S(K,A)$ 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 ${\text{Sha}}_S^2(K,A)$ can increase as $S$ increases to $S\cup X$, and even attain its maximal dimension, ${\dim }_{{\mathbb{F}}_p}{\text{RusB}}_S(K,{\mathbb{F}}_p)$, 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 ${\text{RusB}}_S(K,A)$ to show that ${\text{Sha}}_S^2(K,A)\hookrightarrow {\text{RusB}}_S(K,A)$ and that there exist infinitely many tame sets of primes $X$ of $K$ such that ${\text{Sha}}_{S\cup X}^2(K,A){\text{RusB}}_{S\cup X}(K,A){\text{RusB}}_S(K,A)\hookleftarrow {\text{Sha}}_S^2(K,A)$.