## Student Arithmetic Geometry Seminar: Varieties over a finite field with trivial chow group of $0$-cycles have a rational point.

Seminar | March 15 | 4:10-5 p.m. | 891 Evans Hall

Anningzhe Gao, UCB

Department of Mathematics

We will prove the following statement: If $X$ is a smooth projective variety over a finite field $k$, and the Chow group of $0$-cycles is $\mathbb Z$, then $X$ has a rational point over $k$. We will start from the basic properties of rigid cohomology, then consider a decomposition theorem proved by Spencer Bloch, and finally give the proof by using the trace formula. In particular, the condition on $X$ is satisfied when $X$ is a Fano variety.

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