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Mordell-Weil groups of generic Abelian varieties. (English) Zbl 0576.14020
The paper under review deals with the following question: Suppose \(X=D/\Gamma\) is a Shimura-variety over \(\mathbb C\), and \(A\to X\) a family of abelian varieties derived from the interpretation of \(X\) as a moduli space. What is the Mordell-Weil group \(A(\mathbb C(X))\)? (It is finitely generated.) The author shows that in many cases the group is finite. The details of the proofs are somehow intricate, but the main idea seems to be the following: (a) any rational section extends to a holomorphic section \(X\to A\); (b) over \(X\) we have an exact sequence \[ 0\to\mathcal L\to\mathrm{Lie}(A)\to A\to 0, \] hence an exact sequence \[ H^ 0(X,\mathrm{Lie}(A))\to A(X)\to H^ 1(X,{\mathcal L})\to H^ 1(X,\mathrm{Lie}(A)). \] It suffices to show that the first term vanishes, and that the last arrow is injective up to torsion. If \(X\) is compact we may apply vanishing theorems and Eichler integrals to achieve this. For noncompact \(X\) one reduces to the case of the upper half-space.
Reviewer: Gerd Faltings

MSC:
11G15 Complex multiplication and moduli of abelian varieties
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
14K15 Arithmetic ground fields for abelian varieties
11G18 Arithmetic aspects of modular and Shimura varieties
14K22 Complex multiplication and abelian varieties
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