论文标题
通过专业化来实现阿贝利安品种中的加洛瓦代表
Realizing Galois representations in abelian varieties by specialization
论文作者
论文摘要
我们对以下问题给出了一些积极的答案：给定一个字段$ k $和连续的galois表示$ρ：g_k \ to gl_n（\ Mathbf {q}）$，构建一个小尺寸的Abelian品种$ j/k $ $ρ$，以至于$ρ$是自然$ g_k $ - g_k $ - represprespresprespration bar $ ke $ k $ j（ \ otimes _ {\ Mathbf {Z}} \ Mathbf {q} $。我们证明，如果$ k $具有特征性的$ 2 $，那么对于任何足够大的整数$ g $（取决于$ρ$），我们可以找到许多绝对简单的$ G $ g $二维的Abelian品种，这些品种意识到$ρ$。我们还概述了一种扭曲给定的曲线对称构造的方法，这些曲线具有许多合理的点，而是产生具有大程度的闭合点的曲线，在这种情况下，我们给出了统一的材料构造 - shioda和liu-lorenzini。主要结果是通过应用Néron专业定理的自然概括而获得的。
We give some positive answers to the following problem: Given a field $K$ and a continuous Galois representation $ρ:G_K \to GL_n(\mathbf{Q})$, construct an abelian variety $J/K$ of small dimension such that $ρ$ is a sub-representation of the natural $G_K$-representation on $J(\bar{K}) \otimes_{\mathbf{Z}} \mathbf{Q}$. We prove that if $K$ is Hilbertian of characteristic different from $2$, then for any sufficiently large integer $g$ (depending on $ρ$) we can find infinitely many absolutely simple $g$-dimensional abelian varieties which realize $ρ$. We outline also a method of twisting a given symmetric construction of curves with many rational points to instead produce curves with closed points of large degree, and in this context we give a unified treatment of constructions of Mestre--Shioda and Liu--Lorenzini. The main results are obtained by applying a natural generalization of Néron's Specialization Theorem.
