In order to understand these better, one should look at not just the volume of the period lattice, and the orders of these finite groups or volumes of lattices, but the finite groups, lattices, and endomorphism rings as stuctures themselves. There are refined conjectures (due to Diamond, Flach, and others) about how these groups are related. These involve ``fitting ideals'', representation theory (to make better sense of the -function), etc.

William Stein 2006-10-20