The conjecture involves elliptic curves defined over the rational numbers.

Then there's the pristine white beaches lapped by - it must be said - a chilly Atlantic, their half-moon curves defined by giant boulders.

Neither are algebraic curves or surfaces defined over fields other than the complex numbers.

The modularity theorem, also known as the Taniyama-Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.

As such, algebraic manifolds are a generalisation of the concept of smooth curves and surfaces defined by polynomials.

A compact Riemann surface C that is a connected space is an algebraic curve defined over the complex number field.

However, lathes are constructed by rotating a curve defined by a set of points instead of a function.

In mathematics, the Tate curve is a curve defined over the ring of formal power series with integer coefficients.

The terms implicit curve and implicit surface are usual to denote curves and surfaces defined in this way.

Let be an elliptic curve defined over the finite field , where for a prime and an integer .