双对数和$\mathrm{GL}_3(\mathbb{Q}_p)$的高阶$\mathscr{L}$ -不变量
Dilogarithm and higher $\mathscr{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$
摘要：
设$E$是$\mathbb{Q}_p$的一个充分大的有限扩张，$ρ_p$是一个半稳定表示$\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\rightarrow\mathrm{GL}_3(E)$，它具有秩为2的单值算子$N$和一个非临界Hodge滤子.我们知道$ρ_p$有三个$\mathscr{L}$ -不变量。我们根据$E$中的三个不变量构造了一族$\mathrm{GL}_3(\mathbb{Q}_p)$的局部解析表示，每个不变量都包含由$ρ_p$确定的局部代数表示.当$ρ_p$来自于$G(\mathbb{A}_{\mathbb{Q}_p})$的自守表示$π$对于一个合适的酉群$G_{/\mathbb{Q}}$时，我们证明了在上述族中存在一个唯一的对象嵌入到相应的Heck-同构子空间中的完全上同调中.我们还记得，Breuil构造了一个家庭的本地解析表示依赖于四个不变量，并证明了类似的结果，局部全球兼容性。我们证明，如果一个代表$Π$在Breuil的家庭嵌入到完成上同调，那么它必须同样嵌入到一个对象，在我们的家庭确定的$Π$。这给出了$Π$嵌入完全上同调的一个纯表示论的必要条件。此外，我们的家庭中的每个对象的某些自然subconformants给一个真正的复杂的局部解析表示，实现了派生对象$Σ(λ, \underline{\mathscr{L}})$由Schraen为一个独特的$\underline{\mathscr{L}}$确定的对象。因此，家庭，我们建设之间的关系给出了更高的$\mathscr{L}$ -不变量的研究Breuil和丁和$p$ -adic双对数函数出现在建设$Σ(λ, \underline{\mathscr{L}})$的Schraen。
英文摘要：
Let $E$ be a sufficiently large finite extension of $\mathbb{Q}_p$ and $ρ_p$ be a semi-stable representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\rightarrow\mathrm{GL}_3(E)$ with a rank two monodromy operator $N$ and a non-critical Hodge filtration. We know that $ρ_p$ has three $\mathscr{L}$-invariants. We construct a family of locally analytic representations of $\mathrm{GL}_3(\mathbb{Q}_p)$ depending on three invariants in $E$ with each of them containing the locally algebraic representation determined by $ρ_p$. When $ρ_p$ comes from an automorphic representation $π$ of $G(\mathbb{A}_{\mathbb{Q}_p})$ for a suitable unitary group $G_{/\mathbb{Q}}$, we show that there is a unique object in the above family that embeds into the associated Hecke-isotypic subspace in the completed cohomology. We recall that Breuil constructed a family of locally analytic representations depending on four invariants and proved a similar result of local-global compatibility. We prove that if a representation $Π$ in Breuil's family embeds into the completed cohomology, then it must equally embed into an object in our family determined by $Π$. This gives a purely representation theoretic necessary condition for $Π$ to embed into completed cohomology. Moreover, certain natural subquotients of each object in our family give a true complex of locally analytic representations that realizes the derived object $Σ(λ, \underline{\mathscr{L}})$ by Schraen for a unique $\underline{\mathscr{L}}$ determined by the object. Consequently, the family we construct gives a relation between the higher $\mathscr{L}$-invariants studied by Breuil and Ding and the $p$-adic dilogarithm function which appears in the construction of $Σ(λ, \underline{\mathscr{L}})$ by Schraen.
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