Let F be a totally real field. For a CM quadratic extension K/F, let H_K be the corresponding Hilbert class field. Let \Theta be an infinite collection of CM quadratic extensions of F. For a modular abelian variety A/F, we discuss results on the variation of Mordell-Weil groups A(H_K) as K \in \Theta varies (joint with Ye Tian). Earlier results obtained via varied perspectives go back to Michel-Venkatesh and Templier. The current approach is geometric and seems to differ from the earlier ones.