From the point of view of number theory, the classification of algebraic curves[?] that matters is into three classes, according to their genus g. Suppose given such a curve C defined over the rational numbers (that is, by polynomials with rational coefficients) and non-singular[?] (in this case that condition isn't a real restriction). Then there are three cases, according to how many rational points (points with rational coefficients) are on C
Therefore the conjecture took its natural place in the overall picture.
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