任意数域上PGL（2）的上同调表示的L-不变量
L-invariants for cohomological representations of PGL(2) over arbitrary number fields
摘要：
设$π$是$\mathrm{PGL}_2$在任意签名的数域$F$上的内形式$G$的尖点上同调自守表示.此外，令$\mathfrak{p}$是$F$的素数，使得$G$在$\mathfrak{p}$处分裂，并且$π$在$\mathfrak{p}$处的局部分量$π_\mathfrak{p}$是Steinberg表示。假设该表示在$\mathfrak{p}$是非临界的，我们构造表示$π$的自同构$\mathcal{L}$ -不变量。如果数域$F$是全实的，我们证明了这些自守的$\mathcal{L}$ -不变量与相应的$p$ -adic Galois表示的Fontaine-Mazur $\mathcal{L}$ -不变量一致.这推广了最近的结果Spiess分别Rosso和第一个命名的作者从平行的重量$2$的情况下，任意上同调的重量。
英文摘要：
Let $π$ be a cuspidal, cohomological automorphic representation of an inner form $G$ of $\mathrm{PGL}_2$ over a number field $F$ of arbitrary signature. Further, let $\mathfrak{p}$ be a prime of $F$ such that $G$ is split at $\mathfrak{p}$ and the local component $π_\mathfrak{p}$ of $π$ at $\mathfrak{p}$ is the Steinberg representation. Assuming that the representation is non-critical at $\mathfrak{p}$ we construct automorphic $\mathcal{L}$-invariants for the representation $π$. If the number field $F$ is totally real, we show that these automorphic $\mathcal{L}$-invariants agree with the Fontaine-Mazur $\mathcal{L}$-invariant of the associated $p$-adic Galois representation. This generalizes a recent result of Spiess respectively Rosso and the first named author from the case of parallel weight $2$ to arbitrary cohomological weights.
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