Shafarevich-Tate groups of abelian varieties
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866929343565922304 |
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| author | Nikolaev, Igor V. |
| author_facet | Nikolaev, Igor V. |
| contents | The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (\mathscr{A})\cong Cl~(Λ)\oplus Cl~(Λ)$ or $W (\mathscr{A})\cong \left(\mathbf{Z}/2^k\mathbf{Z}\right) \oplus Cl_{~\mathbf{odd}}~(Λ)\oplus Cl_{~\mathbf{odd}}~(Λ)$, where $Cl~(Λ)$ is the ideal class group of a ring $Λ$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Λ)$. The case of elliptic curves with complex multiplication is considered in detail. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_09970 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Shafarevich-Tate groups of abelian varieties Nikolaev, Igor V. Number Theory Operator Algebras 11G10, 46L85 The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (\mathscr{A})\cong Cl~(Λ)\oplus Cl~(Λ)$ or $W (\mathscr{A})\cong \left(\mathbf{Z}/2^k\mathbf{Z}\right) \oplus Cl_{~\mathbf{odd}}~(Λ)\oplus Cl_{~\mathbf{odd}}~(Λ)$, where $Cl~(Λ)$ is the ideal class group of a ring $Λ$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Λ)$. The case of elliptic curves with complex multiplication is considered in detail. |
| title | Shafarevich-Tate groups of abelian varieties |
| topic | Number Theory Operator Algebras 11G10, 46L85 |
| url | https://arxiv.org/abs/2005.09970 |