@techreport{96e692c6e6064f16b81b22f163da721a, title = "Comparison of different Tate conjectures", abstract = " 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 $A/K$. ", keywords = "math.AG, math.NT, 11G40 (Primary) 11G05, 11G10, 14G10 (Secondary)", author = "Veronika Ertl and Timo Keller and Yanshuai Qin", year = "2024", month = jun, day = "24", doi = "10.48550/arXiv.2012.01337", language = "English", type = "WorkingPaper", }