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Why do we call this group of curves as real and the other one as imaginary? What is the philosophy behind these names?

Why imaginary ones have 1 point at infinity and the real ones have two points at infinity?

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As far as I can tell, this began with E. Artin's thesis, looking at curves over finite fields. The point is that for complex quadratic fields there is just one absolute value (the complex one) extending the archimedean one on the base field $\mathbb Q$ (the real one), while for real quadratic fields there are two. Following the dictum that these extensions should be thought of as "points lying over" produces the terminology for hyperelliptic curves over finite fields.

(Yes, if there were any doubt, in the genuine algebraic-geometry setting, as opposed to the number field setting, the point(s) at infinity can be moved to "finite" positions. For number fields, the "finite" places are certainly not interchangeable with the "infinite" ones.)

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