On the Beilinson-Bloch conjecture over function fields
Matt Broe
TL;DR
The paper extends the Beilinson–Bloch conjecture to global function fields, providing a practical criterion (via the Leray filtration and Frobenius weight arguments) to verify the conjecture in codimension $i$ for a fibration $X\to Y$ whose generic fiber satisfies suitable conditions. It then applies these methods to powers of CM elliptic curves, establishing the Tate conjecture for $E^g\times C$ over finitely generated fields and deriving Beilinson–Bloch conclusions for such products over function fields. The work also develops a detailed CM-endomorphism decomposition of the motive of $E^g$, proves a strong form of Tate for these motives, and computes transcendental motives for $E^2$ and $E^3$, relating Chow groups to Griffiths groups and algebraic equivalence in this CM setting. Overall, the results yield new cases of Beilinson–Bloch over function fields, illuminate the motive-theoretic structure of powers of CM elliptic curves, and address questions about Chow groups in positive characteristic.
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.