论文标题
确定阿贝利亚品种的单肌群
Determining monodromy groups of abelian varieties
论文作者
论文摘要
在一个数字领域，与阿贝里安品种相关的是几个有趣且相关的小组：动机Galois组，Mumford-Tate组，$ \ ell $ - $ - ad-adic monodromomy群组和Sato-Tate组。假设Mumford-Tate的猜想，我们表明，从我们Abelian品种的两个精选的Frobenius多项式中，我们可以恢复这些组的身份成分（或至少是内部形式），直至同构，以及它们的自然表示。我们还通过考虑越来越多的frobenius多项式来获得实用的概率算法来计算这些组。这些组是连接和还原的，因此可以用根数据表表示。这些组与代数周期有猜想的联系，尤其是我们获得了概率算法，以在任何固定程度上计算Abelian品种的Hodge类别的尺寸。
Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree.
