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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2512.03327 |
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| _version_ | 1866908690421907456 |
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| author | Iorga, Andreea Ramakrishna, Ravi |
| author_facet | Iorga, Andreea Ramakrishna, Ravi |
| contents | 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 $\Sha^2_{S\cup X}(K,A) \stackrel{\simeq}{\hookrightarrow} \RusB_{S \cup X}(K,A) \stackrel{\simeq}{\twoheadleftarrow} \RusB_S(K,A) \hookleftarrow \Sha^2_S(K,A)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Increasing the Size of Tame Shafarevich Groups Iorga, Andreea Ramakrishna, Ravi Number Theory 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 $\Sha^2_{S\cup X}(K,A) \stackrel{\simeq}{\hookrightarrow} \RusB_{S \cup X}(K,A) \stackrel{\simeq}{\twoheadleftarrow} \RusB_S(K,A) \hookleftarrow \Sha^2_S(K,A)$. |
| title | Increasing the Size of Tame Shafarevich Groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.03327 |