BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
نویسنده
چکیده
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
منابع مشابه
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
Representations of semisimple Lie algebras in the BGG category í µí²ª, by James E.
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