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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
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What is Mordell-Weil lattice?
[T Shioda :Mordell-Weil lattice][1]
[1]: http://www.rkmath.rikkyo.ac.jp/math/shioda/papers/mwl.pdf more Basic,you also see the homepage of Chao Li about Elliptic Surfaces and Mordell-Weil Lattices …