Classically, an elliptic curve is a connected Riemann surface (a connected compact 1-dimensional complex manifold) of genus 1. The curious term “elliptic” is a remnant from the 19th century, a back-formation which refers to elliptic functions (generalizing circular functions, i.e., the classical trigonometric functions) and their natural domains as Riemann surfaces.
In more modern frameworks, an elliptic curve over a field may be defined as a complete irreducible non-singular algebraic curve of genus 1 over , or even as a certain type of algebraic group scheme. Elliptic curves have many remarkable properties, and their deeper arithmetic study is one of the most profound subjects in present-day mathematics.
Probably should pass through Riemann and Weierstrass, to explain “elliptic”.
Over a general ring
Over the complex numbers
A framed elliptic curve? is an elliptic curve in the sense of the first item in prop. 1, together with an ordered basis of with
A framed lattice in is a lattice together with an ordered basis of such that .
Given an elliptic curve over , , we get a formal group by completing along its identity section
we get a ringed space
If is a field , then the structure sheaf
defines over .
The corresponding formal group law is Euler’s formal group law
if then this is a non-trivial elliptic curve.
If then (additive or multiplicative formal group law corresponding to integral cohomology and K-theory, respectively).
Relation to elliptic cohomology
Elliptic curves, via their formal group laws, give the name to elliptic cohomology theories.
an introduction to elliptic curves is at