Comparison of different Tate conjectures
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866908474051395584 |
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| author | Ertl, Veronika Keller, Timo Qin, Yanshuai |
| author_facet | Ertl, Veronika Keller, Timo Qin, Yanshuai |
| contents | For an abelian variety $A$ over a finitely generated field $K$ of characteristic $p > 0$, we prove that the algebraic rank of $A$ is at most a suitably defined analytic rank. Moreover, we prove that equality, i.e., the BSD rank conjecture, holds for $A/K$ if and only if a suitably defined Tate--Shafarevich group of $A/K$ (1) has finite $\ell$-primary component for some/all $\ell \neq p$, or (2) finite prime-to-$p$ part, or (3) has $p$-primary part of finite exponent, or (4) is of finite exponent. There is an algorithm to verify those conditions for concretely given $A/K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_01337 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Comparison of different Tate conjectures Ertl, Veronika Keller, Timo Qin, Yanshuai Algebraic Geometry Number Theory 11G40 (Primary) 11G05, 11G10, 14G10 (Secondary) For an abelian variety $A$ over a finitely generated field $K$ of characteristic $p > 0$, we prove that the algebraic rank of $A$ is at most a suitably defined analytic rank. Moreover, we prove that equality, i.e., the BSD rank conjecture, holds for $A/K$ if and only if a suitably defined Tate--Shafarevich group of $A/K$ (1) has finite $\ell$-primary component for some/all $\ell \neq p$, or (2) finite prime-to-$p$ part, or (3) has $p$-primary part of finite exponent, or (4) is of finite exponent. There is an algorithm to verify those conditions for concretely given $A/K$. |
| title | Comparison of different Tate conjectures |
| topic | Algebraic Geometry Number Theory 11G40 (Primary) 11G05, 11G10, 14G10 (Secondary) |
| url | https://arxiv.org/abs/2012.01337 |