Titlebook: Number Theory III; Diophantine Geometry Serge Lang Book 1991 Springer-Verlag Berlin Heidelberg 1991 Abelian varieties.Abelian variety.Dimen

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Heights and Rational Points,e domain. Part of determining the solutions consists in estimating the size of such solutions, in various ways. For instance if . are to be elements of the ring of integers Z, then we can estimate the absolute values |x|, |y| or better the maximum max(|x|, |y|). If x, y are taken to be rational numb

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Abelian Varieties,c function associated with every divisor class. Furthermore, the group of rational points can be analyzed as a group, with a description of generators, bounds for the heights of generators, a description of the torsion, all emphasizing the group structure. Thus we collect such results in a separate

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Modular Curves Over ,,ic curves with points of order ., or cyclic subgroups of order .. They form the prototype of higher dimensional versions, modular varieties, which parametrize abelian varieties with other structures involving points of finite order. We have already seen the use of such varieties in Faltings’ proof o

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Arakelov Theory,amounted to the corresponding Riemann surfaces and their differential geometric properties once the number field gets imbedded into the complex numbers. Arakelov showed how one could define a global intersection number for two arithmetic curves on an arithmetic surface, and that this intersection nu

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