关于阿贝尔簇与环面乘积线性相关性的判定问题
On the problem of detecting linear dependence for products of abelian varieties and tori
摘要：
设G是定义在数域K上的一个阿贝尔簇与一个环面的乘积。设R是G（K）中的一点，L是G（K）的一个生成子群.假设对于K的所有素数p，除了p之外，点（R mod p）属于（L mod p）。R是否属于L？我们在三种情况下肯定地回答了这个问题：如果L是循环的;如果L是G（K）的自由左End_K G-子模;如果L有一组生成元（作为一个群）是G（K）的自由左End_K G-子模的基。在一般情况下，我们证明了存在一个整数m（仅依赖于G，K和L的秩）使得mR属于由L生成的G（K）的左End_K G-子模。
英文摘要：
Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a point in G(K) and let L be a finitely generated subgroup of G(K). Suppose that for all but finitely many primes p of K the point (R mod p) belongs to (L mod p). Does it follow that R belongs to L? We answer this question affirmatively in three cases: if L is cyclic; if L is a free left End_K G-submodule of G(K); if L has a set of generators (as a group) which is a basis of a free left End_K G-submodule of G(K). In general we prove that there exists an integer m (depending only on G, K and the rank of L) such that mR belongs to the left End_K G-submodule of G(K) generated by L.
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