While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. In fact, I've even asked people, who have probably given me perfectly good answers, but somehow their answers have never stuck in my brain. The question is simple: the Poisson summation formula is incredibly useful to many people, but why is that? When you first see it, it looks like a piece of magic, but then suddenly you start spotting that people keep saying "By Poisson summation" and expecting you to fill in the details. In that respect, it's a bit like the phrase "By compactness," but the important difference for me is that I can fill in the details of compactness arguments.
What I would like to know is this. What is the "trigger" that makes people think, "Ah, Poisson summation should be useful here"? And is there some very simple example of how it is applied, with the property that once you understand that example, you basically understand how to apply it in general? (Perhaps two or three examples are needed -- that would obviously be OK too.) And can one give a general description of the circumstances where it is useful? (Anyone familiar with the Tricki will see that I am basically asking for a Tricki article on the formula. But I don't mind something incomplete or less polished.)
For reference, here is a related (but different) question about the Poisson summation formula: Truth of the Poisson summation formula