As a natural generalization of the notion of `higher rank Euler system', we develop a theory of `higher special elements' in the exterior power biduals of the Galois cohomology of $p$-adic representations. We show, in particular, that such elements encode detailed information about the structure of Galois cohomology groups and are related by families of congruences involving natural height pairings on cohomology. As a first concrete application of the approach, we use it to refine, and extend, a variety of existing results and conjectures concerning the values of derivatives of Dirichlet $L$-series.

中文翻译：

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关于 p-adic 表示的高级特殊元素

作为“高阶欧拉系统”概念的自然概括，我们在 $p$-adic 表示的伽罗瓦上同调的外部幂二元中发展了“更高特殊元素”的理论。我们特别表明，这些元素编码关于伽罗华上同调群结构的详细信息，并且与涉及上同调上自然高度配对的同余族相关。作为该方法的第一个具体应用，我们使用它来改进和扩展有关 Dirichlet $L$ 系列导数值的各种现有结果和猜想。