I know the book. And it starts with a discussion on the question of whether integral solutions of homogeneous systems are morally the same as rational points on the underlying projective variety, an interesting and puzzling point (until you realise that the issue must be about parametric families of solutions). I gave a historical answer to a methodological question, which may seem perverse. But "history written by the victors" plays a major role in talking about methods.

That's being pedantic - I know what was meant and it was the Langlands program. And/or the Antwerp Modular Forms Conference 1972. The intellectual consequences of Borel and Serre's involvement in the Seminar on Complex Multiplication.

Indeed, but the article says there is more than one definition. Which implies that the concept is used somewhat like "fractal": in a science text it may mean something but what it means may not be a piece of mathematics.

en.wikipedia.org/wiki/Quasicrystal. Please try Wikipedia before posting here.

Well, it's a matter of paraphrase, then. The second para of your question suggests you would like something more than just a sufficient condition on the topology. Taking the complement of a point in three-space, the topologist might divide it into two pieces that were contractible and overlapped.

Yes, there's the good point I missed that where the numbers are rationally dependent there is the geometry explained in another answer.

My answer has a definition of ordinary but not of supersingular, on the other hand.

It's great to see this site working in a way that is close to actual research.