摘要翻译：
在D.Ramakrishnan之后，我们解释了L.Lafforgue的模性定理和H.Jacquet和J.Shalika的解析定理如何被应用于证明与Tate猜想有关的以下结果：对于定义在全局函数域上的光滑的、射影的、几何连通的簇，代数秩小于或等于解析秩。本文还讨论了数域的类似（开放）问题和Lafforgue定理的一个简单推广，以消除“有限阶特征”假设。所有的结果都可能“为专家所知”，但似乎没有被写下来。
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英文标题：
《A rank inequality for the Tate Conjecture over global function fields》
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作者：
Christopher Lyons
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最新提交年份：
2015
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分类信息：

一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank.   Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be written down.
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PDF链接：
https://arxiv.org/pdf/0812.0094