Suppose we are given an algebraic curve C defined over the rational numbers (that is, C is defined by polynomials with rational coefficients), and suppose further that C is non-singular (though in this case that condition isn't a real restriction). How many rational points (points with rational coefficients) are on C?
The answer depends upon the genus g of the curve. As is common in number theory, there are three cases: g = 0, g = 1, and g greater than 1. The g = 0 case has been understood for a long time; Mordell solved the g = 1 case, and conjected the result for the g greater than 1 case.
The complete result is this:
Let C be an non-singular algebraic curve over the rationals of genus g. Then the number of rational points on C may be determined as follows: