Classically in complex geometry, an elliptic curve is a connected Riemann surface (a connected compact 1-dimensional complex manifold) of genus 1, hence it is a torus equipped with the structure of a complex manifold, or equivalently with conformal structure.
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 and in the generality of algebraic geometry, an elliptic curve over a field or indeed over any commutative ring may be defined as a complete irreducible non-singular algebraic curve of arithmetic 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.
The moduli stack of elliptic curves equipped with its canonical map to the moduli stack of formal group laws plays a central role in chromatic homotopy theory at chromatic level 2, where it serves to parameterize elliptic cohomology theories.
Elliptic curves over the complex numbers are also interpreted as those worldsheets in string theory whose correlators are the superstring’s partition function, which is the Witten genus. Via the string orientation of tmf this connects to to the role of elliptic curves in elliptic cohomology theory.
Elliptic curves over a general commutative ring (hence in arithmetic geometry) are the well-behaved 1-dimensional group objects parameterized over the space (the prime spectrum of ). (Notice the count of dimension: over the complex numbers a torus is complex 1-dimensional and in this sense one is looking at 1-dimensional group schemes here.) This we discuss below in
An elliptic curve over a field of positive characteristic whose formal group law has height equal to 2 is called a supersingular elliptic curve. Otherwise the height equals 1 and the elliptic curve is called ordinary.
Elliptic curves are examples of solutions to Diophantine equations of degree 3. We start by giving the equation valued over general rings, which is fairly complicated compared to the special case that it reduces to in the classical case over the complex numbers. The more elements in the ground ring are invertible, the more the equation may be simplified.
Much of the literature on elliptic curves considers def. 3 for the case that is an algebraically closed field, in which case there is no need to pass to a cover. But for the true global discussion necessary for the moduli stack of elliptic curves one needs the full generality.
The non-singular such solutions are the elliptic curves over . Non-singularity is embodied in coordinates as follows.
and in terms of these
Here is called the discriminant.
called the j-invariant.
In def. 4 the discriminant satisfies the relation
The following is a definition if one takes the coordinate-description as fundamental. If one takes the more abstract characterization of def. 1 as fundamental then the following is a proposition.
If moreover is also invertible in , hence generally over then this equation is equivalent to just
If the ring is the complex numbers, then complex tori are indeed the solutions to the Weierstrass equation as in prop. 1, parameterized by a torus (as discussed in the section in terms of complex geometry) via the Weierstrass elliptic function as in the form
See e.g. (Hain 08, section 5) on how complex elliptic curves are expressed in this algebraic geometric fashion.
A framed lattice in is a lattice together with an ordered basis of such that .
Hence a framed elliptic curve is the quotient of the complex plane by a lattice together with the information on how this quotient was obtained. This is useful for describing the moduli stack of elliptic curves over the complex numbers.
In the case over the complex numbers an elliptic curve is equivalently the quotient of the complex plane by a framed lattice. If here one remembers the structure given by that framed lattice, this means equivalently to remember an ordered basis
If here one replaces the integers by a cyclic group then one obtains what is called a level-n structure on an elliptic curve. Level- structures on elliptic curves may also be defined over general rings.
These structures are useful in that the moduli stack of elliptic curves with level-n structure (a modular curve in the case over the complex numbers) provides a finite covering of the full moduli stack of elliptic curves.
we get a ringed space
defines over .
The corresponding formal group law is Euler’s formal group law
if then this is a non-trivial elliptic curve.
Classical accounts of the general case include
Nicholas M. Katz, Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR MR772569 (86i:11024)
Introductory lecture notes for elliptic curves over the complex numbers include
Richard Hain, Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)
and for the general case
A quick introduction to elliptic curves (pdf)
R. Sujahta, Elliptic Curves & Number Theory (pdf)
For more along these lines see also at arithmetic geometry.
In the context of elliptic fibrations:
A general textbook account is
Discussion over the rational numbers includes
Discussion of elliptic curves over the p-adic numbers includes
Rosa Winter, Elliptic curves over , 2011 (pdf)