Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups
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arXiv
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| Format: | Preprint |
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2026
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| author | Čoupek, Pavel Gazaki, Evangelia Marmora, Adriano |
| author_facet | Čoupek, Pavel Gazaki, Evangelia Marmora, Adriano |
| contents | Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{Gal}(\overline{K}/K)$ with Hodge-Tate weights in $[0, r]$. Assuming $0\leq r<p-1$, we prove that for a semistable extension of $\mathbb{Z}_p$ by $T$, the corresponding sequence of strongly divisible modules is exact. The same statement is proved for Breuil-Kisin modules for all $r\geq 0$. In the crystalline case, we deduce that the integral Bloch-Kato Selmer group $H^1_f(K, T)$ is computed by $\mathrm{Ext}^1$ in the category of crystalline strongly divisible modules. Using further exactness results, we define a tensor product of strongly divisible modules, which commutes with the functors to Galois representations. As an application, we show that for abelian varieties $A_1, A_2$ over $K$ with good reduction, the cup product map $δ_1\cupδ_2:A_1(K)\otimes A_2(K)\rightarrow H^2(K, T_p(A_1)\otimes T_p(A_2))$ induced by the Kummer sequences of $A_1, A_2$ factors through an $\mathrm{Ext}^2$ group of strongly divisible modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26035 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups Čoupek, Pavel Gazaki, Evangelia Marmora, Adriano Number Theory Algebraic Geometry 11S25, 11F80, 14F20, 14F30 Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{Gal}(\overline{K}/K)$ with Hodge-Tate weights in $[0, r]$. Assuming $0\leq r<p-1$, we prove that for a semistable extension of $\mathbb{Z}_p$ by $T$, the corresponding sequence of strongly divisible modules is exact. The same statement is proved for Breuil-Kisin modules for all $r\geq 0$. In the crystalline case, we deduce that the integral Bloch-Kato Selmer group $H^1_f(K, T)$ is computed by $\mathrm{Ext}^1$ in the category of crystalline strongly divisible modules. Using further exactness results, we define a tensor product of strongly divisible modules, which commutes with the functors to Galois representations. As an application, we show that for abelian varieties $A_1, A_2$ over $K$ with good reduction, the cup product map $δ_1\cupδ_2:A_1(K)\otimes A_2(K)\rightarrow H^2(K, T_p(A_1)\otimes T_p(A_2))$ induced by the Kummer sequences of $A_1, A_2$ factors through an $\mathrm{Ext}^2$ group of strongly divisible modules. |
| title | Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups |
| topic | Number Theory Algebraic Geometry 11S25, 11F80, 14F20, 14F30 |
| url | https://arxiv.org/abs/2603.26035 |