On generators of the Chow group of 0-cycles on diagonal cubic surfaces over 3-adic fields
We show that the degree-zero part of the Chow group of $0$-cycles on some diagonal cubic surface over a 3-adic field is generated by the classes of rational points. In the proof of the main result, we will construct explicit generators over the cubic unramified scalar extension. A key point is to check that such cycles have non-zero values under the Brauer–Manin pairing. We will use the Hilbert symbol to compute them.