The purpose of this article is proving the equality of two natural $\mathcal{L}$‐invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic $\mathcal{L}$‐invariant as a universal norm of a canonical ${\mathbb{Z}}_p$‐extension of ${\mathbb{Q}}_p$ associated to the representation. We relate it to a certain $2\times 2$ regulator of $p$‐adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic $\mathcal{L}$‐invariant computed in Rivero and Rotger [J. Eur. Math. Soc., to appear].

中文翻译：

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关于重的一种模形式的伴随的L不变量

本文的目的是证明两个自然 $\mathcal{大号}$-附在一个权重尖点形式的伴随表示上的不变量，每种形式分别由纯解析的代数方法定义。证明与格林伯格对代数的定义背道而驰$\mathcal{大号}$不变式为规范的通用规范 ${\mathbb{ž}}_p$-的扩展 ${\mathbb{问}}_p$与表示相关联。我们把它与某个$2\times 2$ 监管者 $p$-通过类场理论得出的全球单位的对数对数，然后我们证明它等于解析 $\mathcal{大号}$在Rivero和Rotger中计算的不变量[ J. Eur。数学。Soc。，出现]。