arXiv subjects

Matt Broe

Publications and source records attributed to Matt Broe.

3 recordsLinked to original sources

arXiv2026

The Tate conjecture for abelian fourfolds over finite fields

We prove the Tate conjecture for abelian fourfolds over finite fields. This is the first resolution of the conjecture for all abelian varieties of a fixed dimension over finite fields since the work of Tate in the 1960s. The proof relies on techniques from Ancona's proof of the standard conjecture of Hodge type for abelian fourfolds, and ultimately reduces to Markman's results on the algebraicity of Weil classes on complex abelian varieties. Combining the above case of the Tate conjecture with theorems of Ancona and Kahn, we deduce that the standard conjecture on homological versus numerical equivalence holds for abelian fourfolds over arbitrary fields. This completes the proof of the standard conjectures for abelian fourfolds.

math.AG↗

arXiv2025

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

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.

math.NT↗

arXiv2025

On the Beilinson-Bloch conjecture over function fields

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.

math.NT↗