So what do I mean by "well-connected"? Elliptic curves are associated with
a) number theory - many questions about number theory boil down to finding points on elliptic curves with rational coordinates
b) algebraic geometry - elliptic curves are very "nice" from this point of view, and they are complicated enough that you can say interesting things, but not so complicated that you can't say anything
c) complex analysis - over the complex plane, an elliptic curve is a torus. You might have seen how a torus can be formed by identifying opposite edges of a square, and an elliptic curve is hence "a quotient of C by a square lattice". Modular forms, which are creatures of complex analysis with deep applications to number theory, are naturally defined in this framework.
and many more things I don't know about. They're in a sort of "sweet spot" and act as a bridge between multiple parts of mathematics.
a) number theory - many questions about number theory boil down to finding points on elliptic curves with rational coordinates
b) algebraic geometry - elliptic curves are very "nice" from this point of view, and they are complicated enough that you can say interesting things, but not so complicated that you can't say anything
c) complex analysis - over the complex plane, an elliptic curve is a torus. You might have seen how a torus can be formed by identifying opposite edges of a square, and an elliptic curve is hence "a quotient of C by a square lattice". Modular forms, which are creatures of complex analysis with deep applications to number theory, are naturally defined in this framework.
and many more things I don't know about. They're in a sort of "sweet spot" and act as a bridge between multiple parts of mathematics.