Let S be a certain affine algebraic surface over \({\mathbb Q}\) such that it admits a regular map to \({\mathbb A}^2/{\mathbb Q}\) . We show that any non-trivial torsion element in the Chow group \({ {CH}}^1(S)\) can be pulled back to ideal classes of quadratic fields whose order can be made as large as possible. This gives an affirmative answer to a question analogous to one raised by Agboola and Pappas, in the case of certain affine algebraic surfaces. Spreading out S over \({\mathbb Z}\) and for a closed point \(P\in {\mathbb A}^2/{\mathbb Z}\) , we show that the cardinality of a subgroup of the Picard group of the fiber \(S_P\) remains unchanged when P varies over a Zariski open subset in \({\mathbb A}^2/{\mathbb Z}\) . We also show by constructing an element of odd order \(n\ge 3\) in the class group of certain imaginary quadratic fields that the Picard group of \(S_P\) has a subgroup isomorphic to \({\mathbb Z}/n{\mathbb Z}\) .
