I am a math undergrad, so much of the literature on elliptic curves escapes me. I'm trying to understand why one considers elliptic curves over the complex numbers. Specifically, this part of the Wikipedia article is very technical and hard to understand for me. Silverman and Tate say the following:
In general, two cubic curves meet in nine points. To make that statement correct, one should first of all use the projective plane, which has extra points at infinity. Secondly, one should introduce multiplicities of intersections, counting points of tangency for example as intersections of multiplicity greater than one. And finally, one must allow complex numbers for coordinates. [emphasis mine]
I understand the two first musts, but why must we allow complex numbers for coordinates? An explanation in language understandable by an undergrad would be appreciated. Thanks.