论文标题
大型主要系列，p-adic家庭和l-invariants
Big principal series, p-adic families and L-invariants
论文作者
论文摘要
在较早的工作中，第一个命名的作者概括了达尔蒙风格的$ \ mathcal {l} $ - 不变的cuspidal自动形态表示，该组的高级半精子群是相对于琐碎系数系统和Steinberg的共同学。 在本文中，假设该组的Archimedean组成部分具有离散的系列，我们表明这些自动形态$ \ Mathcal {l} $ - 可以根据$ p $ - 亚种家庭中的hecke-eigenvalues的衍生物来计算不变性。即使在模块化形式的情况下，我们的证明也是新颖的，而模块化形式是由Bertolini，Darmon和Iovita建立的。主要的新技术成分是Kohlhaase和Schraen的局部分析主序列表示的Koszul分辨率。 作为结果的应用，我们解决了Spieß的猜想：我们表明自动形态$ \ Mathcal {l} $ - Hilbert模块化的平行重量$ 2 $不变的模块化形式$ 2 $独立于用于定义它们的符号。此外，我们表明它们在Jacquet-Langlands转移下是不变的，实际上等于Fontaine-Mazur $ \ Mathcal {l} $ - 相关Galois代表的不变性。在温和的假设下，我们还证明了自动形态和Fontaine-Mazur $ \ Mathcal {l} $的平等性 - 不变的，用于确定的任意等级的确定统一群体的表示。 最后，我们研究了Bianchi模块化形式的情况，以说明在没有离散串联表示的情况下，我们的方法（给定特征变量的精确结果）如何工作。
In earlier work, the first named author generalized the construction of Darmon-style $\mathcal{L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal{L}$-invariants can be computed in terms of derivatives of Hecke-eigenvalues in $p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon, and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic $\mathcal{L}$-invariants of Hilbert modular forms of parallel weight $2$ are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet-Langlands transfer and, in fact, equal to the Fontaine-Mazur $\mathcal{L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine-Mazur $\mathcal{L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.
