Elliptic curves over finite fields have immediate applications to cryptography, but elliptic curves over the complex numbers are cool too. Just like the sine an

Elliptic curves over finite fields have immediate applications to cryptography, but elliptic curves over the complex numbers are cool too. Just like the sine and cosine parametrize the circle, elliptic functions parametrize elliptic curves. Whereas trig functions are singly periodic with period 2pi, elliptic functions are doubly periodic functions, meaning that there are fixed complex numbers z and w such that f(x+z) = f(x+w) = f(x). You can prove that for a given period lattice, the field of such functions is generated by two such fundamental functions—-the Weierstrass elliptic function and its derivative. The addition law for elliptic curves over C can be derived from something like the equivalent of the addition law for sine and cosine. Elliptic functions have applications to physics and are also interesting in their own right. This all leads to the study of fractional linear transformations, modular forms and even Fermat’s Last Theorem. It’s all quite lovely.