Strong BSD for abelian surfaces and the Bloch-Beilinson conjecture

TL;DR

The paper proves a finiteness result for zero-cycles on abelian surfaces over number fields, establishing that $A_0(A_K)\otimes{\mathbb{Q}}$ is finite-dimensional and relating its rank to $\operatorname{ord}_{s=2}L(H^3(A),s)$. It connects this finiteness to the Birch–Swinnerton-Dyer conjecture for abelian surfaces, showing that, over $\mathbb{Q}$, strong BSD implies Bloch–Beilinson for $A_0(A_{\mathbb{Q}})$, and that vanishing of $L(A,1)$ forces the zero of $L(H^3(A),s)$ at $s=2$ to be nonzero via Poincaré duality. The work further leverages Roitman’s torsion theorem to prove Albanese kernel triviality over $\bar{K}$ and discusses implications for Kummer surfaces, including $A_0(X)=0$ for the associated Kummer K3 surface $X$. Overall, the results provide a pathway from BSD to Bloch–Beilinson for abelian surfaces and illuminate the structure of zero-cycles via $L$-functions and Albanese kernels.

Abstract

In this paper, we prove the Bloch-Beilinson conjecture for certain abelian surfaces over $\mathbb{Q}$, provided that the BSD is known for these abelian surfaces.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2