BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

Using a geometric approach turns out to be extremely fruitful. There are several simplifications which one can make. First, if the polynomial f(X,Y ) factors in C[X,Y ], then C breaks up into a union of two or more curves, each of which may be studied separately, so it suffices to study curves defined by irreducible polynomials. Second, the curve C may be singular, but we can always replace C with a non-singular curve C̃ so that there is a map C̃ → C which is bijective at all but finitely many points. It thus suffices to study integral and rational points on non-singular curves. Third, we can embed a non-singular curve C into a complete non-singular curve C̄ so that the complement C̄rC consists of a finite set of points. It turns out that the geometry of C̄ and C̄ r C largely determines the qualitative behavior of the integral and rational points on C. Let K be a number field (e.g. Q), let R be a finitely generated subring of K (e.g. Z), and let C be a non-singular curve defined by polynomial equations with coefficients in R. As above, we will let C̄ be a non-singular completion of C. The complex points of C̄, denoted C̄(C), form a Riemann surface, and the complement C̄(C)rC(C) consists of a finite (possibly empty) set of points. The Euler characteristic of C, denoted χ(C), can be defined as the usual alternating sum of vertices, edges, and faces of a triangularization of C(C), or by using any one of the standard (co)homology theories. Equivalently, if we write g(C̄) for the genus of the Riemann surface C̄(C), then