On the Beilinson-Bloch conjecture over function fields

TL;DR

This paper extends the Beilinson-Bloch conjecture to function fields, providing criteria, new proofs, and special case verifications, linking several major conjectures in algebraic geometry and number theory.

Contribution

It generalizes the Beilinson-Bloch conjecture to global function fields and proves the Tate conjecture for certain products involving CM elliptic curves.

Findings

Established a criterion for the Beilinson-Bloch conjecture over function fields.

Provided a new proof connecting Tate and Birch–Swinnerton-Dyer conjectures.

Proved the Tate conjecture for products of curves and CM elliptic curves.

Abstract

Let $k$ be a field and $X$ a smooth projective variety over $k$. When $k$ is a number field, the Beilinson-Bloch conjecture relates the ranks of the Chow groups of $X$ to the order of vanishing of certain $L$-functions. We consider the same conjecture when $k$ is a global function field, and give a criterion for the conjecture to hold for $X$, extending an earlier result of Jannsen. As an application, we provide a new proof of a theorem of Geisser connecting the Tate conjecture over finite fields and the Birch and Swinnerton-Dyer conjecture over function fields. We then prove the Tate conjecture for a product of a smooth projective curve with a power of a CM elliptic curve over any finitely generated field, and thus deduce special cases of the Beilinson--Bloch conjecture. In the process we obtain a conditional answer to a question of Moonen on the Chow groups of powers of ordinary CM elliptic curves over arbitrary fields.