Abstract
We prove a finiteness result on the torsion subgroup in the Chow group of zero cycles on the elliptic modular surface of level four. The main ingredient is Shioda's interpretation of this surface as the Kummer surface associated to the self-product of a certain elliptic curve. On the way we extend the main finiteness theorem on torsion zero cycles on the self-product of a modular elliptic curve to the case where the elliptic curve has complex multiplication and its conductor is a power of a prime.
