The Beilinson-Bloch conjecture for some non-isotrivial varieties over global function fields

Abstract

The Beilinson--Bloch conjecture is a generalization of the Birch and Swinnerton-Dyer conjecture, which relates the ranks of Chow groups of smooth projective varieties over global fields to the order of vanishing of $L$-functions. We prove the conjecture for certain classes of non-isotrivial varieties over $\mathbb{F}_q(t)$, including some cubic threefolds and fivefolds. We deduce the Birch and Swinnerton-Dyer conjecture for their intermediate Jacobians, and use it to establish new cases of the Tate conjecture over finite fields. We also prove further results on the arithmetic of these intermediate Jacobians. To that end, we show that a few classes of varieties over an arbitrary field have motive of abelian type, generalizing previously known examples over the complex numbers.