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arXiv 1708.07899math.NT
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阿贝尔簇的一个Radical刻画

A Radical Characterization of Abelian Varieties

Theodore Hui

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摘要：

设$A$是定义在数域$K$上的无平方阿贝尔簇。设$S$是$\mathcal{O}_K$的素理想$\mathfrak{p}$的密度为1的集合.一个著名的定理说，Frobenius多项式$P_{A,\mathfrak{p}}(x)$为$\mathfrak{p}\in S$确定$A$，直到isk。我们证明了$|A(\mathbb{F}_{\mathfrak{p}})|=P_{A,\mathfrak{p}}(1)$对于$\mathfrak{p}\in S$的素因子也决定了$A$直到在$K$的显式有限扩张上的等距。证明依赖于理解来自$A$的$\ell$ -adic Galois表示的$\ell$ -adic单值群，以及绝对Weyl群对它们的权的作用。

英文摘要：

Let $A$ be a square-free abelian variety defined over a number field $K$. Let $S$ be a density one set of prime ideals $\mathfrak{p}$ of $\mathcal{O}_K$. A famous theorem of Faltings says that the Frobenius polynomials $P_{A,\mathfrak{p}}(x)$ for $\mathfrak{p}\in S$ determine $A$ up to isogeny. We show that the prime factors of $|A(\mathbb{F}_{\mathfrak{p}})|=P_{A,\mathfrak{p}}(1)$ for $\mathfrak{p}\in S$ also determine $A$ up to isogeny over an explicit finite extension of $K$. The proof relies on understanding the $\ell$-adic monodromy groups which come from the $\ell$-adic Galois representations of $A$, and the absolute Weyl group action on their weights.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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