TITLE: Examples and intuition for arithmetic schemes
QUESTION [13 upvotes]: How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good references? What kind of intuition do people have for such schemes?

REPLY [3 votes]: One example I always found useful was that if you consider an elliptic curve like (the projective model of) y^2=x^3+1, then this equation gives an elliptic curve not only over the complex numbers but over any field of characteristic not 2 or 3. So in fact it's giving an elliptic curve over Spec(Z[1/6]), which will look like a scheme with a map to Spec(Z[1/6]) such that the fibre over the point corresponding to the prime ideal p>3 is just the curve over Z/pZ, and the point corresponding to the fibre over the generic point (0) is just the curve over Q. I think that checking these statements formally gave me some sort of intuition as to what was going on at the time. Note that Spec(Z[1/6]) is just the open subscheme of Spec(Z) that you get by throwing away the two closed points (2) and (3).