Cohomology of fiber systems and Mordell-Weil groups of abelian varieties. (English) Zbl 0645.14019
Given a real semisimple Lie group \(G\) with an arithmetic subgroup \(\Gamma\) and a representation of \(G\) in a symplectic vector space \(V\), one obtains a fiber system of abelian varieties over \(\Delta =D/\Gamma\), where \(D\) is the Hermitian symmetric domain associated to \(G\). The generic fiber being denoted by \(A\), it is shown that the Mordell-Weil group \(A(\mathbb C(\Delta))\) is finite if \(H^0(\Gamma,V)\) and \(H^1(\Gamma,V)\) are trivial.
Reviewer: J. H. de Boer
MSC:
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11F75 | Cohomology of arithmetic groups |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14K10 | Algebraic moduli of abelian varieties, classification |
| 22E40 | Discrete subgroups of Lie groups |
| 22E46 | Semisimple Lie groups and their representations |
| 32L10 | Sheaves and cohomology of sections of holomorphic vector bundles, general results |
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