GL（n）的泛非分歧模与Ihara猜想
The universal unramified module for GL(n) and the Ihara conjecture
摘要：
设$F$是$\mathbb{Q}_p$的有限扩张。设$W(k)$表示特征$\ell$不同于$p$和$2$的代数闭域$k$的Witt向量，$\mathcal{Z}$表示$W(k)$上$GL_n(F)$的球面Hecke代数.给定一个Hecke特征标$λ:\mathcal{Z}\to R$，其中$R$是一个任意的$W(k)$ -代数，我们引入了泛非分歧模$\mathcal{M}_{λ,R}$ .证明了$\mathcal{M}_{λ,R}$嵌入其Whittaker空间，并且在$R$上是平坦的，解决了Lazarus的一个猜想.因此，$\mathcal{M}_{λ,k}$具有与任何具有Hecke特征标$λ$的非分歧主级数相同的半化简性。在mod- $\ell$自守形式的背景下，Clozel、Harris和Taylor提出了一个类似于Ihara引理的数学类比。它预言了mod- $\ell$自守形式的循环模$V$的每一个不可约子模都是泛型的。我们的结果惠特克模型的$\mathcal{M}_{λ,k}$减少了Ihara猜想的声明，$V$是通用的。
英文摘要：
Let $F$ be a finite extension of $\mathbb{Q}_p$. Let $W(k)$ denote the Witt vectors of an algebraically closed field $k$ of characteristic $\ell$ different from $p$ and $2$, and let $\mathcal{Z}$ be the spherical Hecke algebra for $GL_n(F)$ over $W(k)$. Given a Hecke character $λ:\mathcal{Z}\to R$, where $R$ is an arbitrary $W(k)$-algebra, we introduce the universal unramified module $\mathcal{M}_{λ,R}$. We show $\mathcal{M}_{λ,R}$ embeds in its Whittaker space and is flat over $R$, resolving a conjecture of Lazarus. It follows that $\mathcal{M}_{λ,k}$ has the same semisimplification as any unramified principle series with Hecke character $λ$. In the setting of mod-$\ell$ automorphic forms, Clozel, Harris, and Taylor formulate a conjectural analogue of Ihara's lemma. It predicts that every irreducible submodule of a certain cyclic module $V$ of mod-$\ell$ automorphic forms is generic. Our result on the Whittaker model of $\mathcal{M}_{λ,k}$ reduces the Ihara conjecture to the statement that $V$ is generic.
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