北京雁栖湖应用数学研究院

刁晗生 , 胡悦科 , 埃马纽埃尔·勒库图里耶 , 凯撒·鲁普

Ziquan Yang

2022年09月20日 10:00 至 11:00

1118

Zoom 293 812 9202 (BIMSA)

The past decade has witnessed a great advancement on the Tate conjecture for varietes with Hodge number $h^{2,0}=1$. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary $h^{2,0}=1$ varietes in characteristic $0$. In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective $h^{2,0}=1$ when $p>>0$, under a mild assumption on moduli. By refining this general result, we prove that in characteristic $p\ge 5$ the BSD conjecture holds true for height $1$ elliptic curve $\mathcal{E}$ over a function field of genus $1$, as long as $\mathcal{E}$ is subject to the generic condition that all singular fibers in its minimal compactification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz $(1,1)$-theorem over $\mathbb{C}$ is very robust for $h^{2,0}=1$ varietes, and works well beyond the hyperkahler world. This is a joint work with Paul Hamacher and Xiaolei Zhao.