Comparison of different Tate conjectures

authored by

Veronika Ertl, Timo Keller, Yanshuai Qin

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 $A/K$.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Preprint

Publication date

24.06.2024

Publication status

E-pub ahead of print

Electronic version(s)

https://doi.org/10.48550/arXiv.2012.01337 (Access: Open)