I've just glanced (fast) through the problem book and I wanted to send you a note telling you that I think it is wonderful. It really seems enticing, rich, friendly, and vastly interesting. About the 144169 problem, there was a bit of discussion about it and related things last Spring (Kevin Buzzard, and Robert Pollack, in particular, had ideas). The more general question behind the 144169 example is to consider the p-adic (Hida) Hecke algebra (say, of tame level 1, for starters) as a -algebra and to form D = discriminant of the finite flat -algebra , so that is in (i.e., is an Iwasawa function). We want to know something about the basic invariants of , e.g., its " -invariant " in each of the discs that form the rigid-analytic space underlying and more specifically, we want to know something about the placement of the zeroes of , if there are any. With p=144169 and the 24th disc, since that part of is quadratic over that part of , there could be some zeroes, so the question is: are there some, and how many? If I remember right, Kevin had an idea about how to quickly compute this and Pollack had an idea of how--in the context of some tame level--to get examples for low primes like and , where has some zeroes.