TY - UNPB

T1 - Comparison of different Tate conjectures

AU - Ertl, Veronika

AU - Keller, Timo

AU - Qin, Yanshuai

PY - 2024/6/24

Y1 - 2024/6/24

N2 - 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 $A/K$.

AB - 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 $A/K$.

KW - math.AG

KW - math.NT

KW - 11G40 (Primary) 11G05, 11G10, 14G10 (Secondary)

U2 - 10.48550/arXiv.2012.01337

DO - 10.48550/arXiv.2012.01337

M3 - Preprint

BT - Comparison of different Tate conjectures

ER -