$p$-adic $L$-functions and unitary completions of representations of $p$-adic reductive groups
$p$-adic $L$-functions and unitary completions of representations of $p$-adic reductive groups
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$p$-adic $L$-$p$-adic 还原群表示的函数和酉完成
DOI：
10.1215/00127094-8230018
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发表时间：
2005
影响因子：
2.5
通讯作者：
M. Emerton
中科院分区：
文献类型：
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作者：
M. Emerton
在本文的前半部分，我们引入了局部凸p进向量空间上p进约群连续表示的普遍酉补的概念，并在适当的假设下证明了这种补全的存在。Breuil在研究GL2的可能的“p进局部朗兰对应”时提出了研究酉补的问题，我们将我们的构造与Breuil对群GL2（Qp）的某些猜想联系起来。特别地，我们证明了局部代数特征的局部解析抛物归纳的普遍酉补与相应的局部代数归纳的普遍酉补是一致的，只要所归纳的特征满足“非临界斜率”条件。（见下文提案2.5）在论文的第二部分，我们考虑了经典模曲线的p-径向补全上同调所得到的GL2（Qp）的某种幺正Banach空间表示。该表示的存在意味着由经典有限斜率新形式产生的GL2（Qp）的局部代数抛物诱导表示具有非平凡的普遍酉补性（验证了这些表示的Breuil猜想），而在此背景下应用命题2.5使我们能够给出附加于p稳定非临界斜率新形式的p进l函数的新构造。结合我们的构造和基于Breuil[5]的l -不变量的表示理论定义，我们能够给出Mazur-Tate-Teitelbaum例外零猜想的简单证明（根据Breuil的l -不变量定义）。这张便条的目的是双重的。在它的前半部分，我们考虑了p进群的表示理论中的以下问题：在p进拓扑向量空间上的p进约简群G的拓扑不可约连续表示（可能满足一些附加假设）何时允许嵌入到p进巴纳赫空间上G的酉表示中？如果G = GL2（Qp）并且所考虑的表示是可容许的局部代数，这个问题已经由Breuil在他关于“GL2（Qp）的p进局部Langlands对应”的思想中提出[4,5]。在论文的第二部分，我们解释了一个关于模形式的p进l函数的表示理论观点。作为一个应用，我们证明了Mazur-Tate-Teitelbaum例外零猜想的一个版本[16，第46页]，使用Breuil在[5]中给出的l -不变量的表示理论定义（它提供了Darmon[9]和Orton[18]分别在权值2和更高权值情况下给出的定义的重新解释）。论文的两部分比乍一看更密切相关。事实上，前半部分讨论的本地工具在本文中发挥了至关重要的作用。作者要感谢美国国家科学基金会（奖励号DMS-0401545）的支持。
In the first half of the paper we introduce the notion of the universal unitary completion of a continuous representation of a p-adic reductive group on a locally convex p-adic vector space, and prove that such a completion exists under appropriate hypotheses. The problem of studying unitary completions has been raised by Breuil in connection with his work on a possible “p-adic local Langlands correspondence” for GL2, and we relate our construction to certain conjectures of Breuil for the group GL2(Qp). In particular, we show that the universal unitary completion of the locally analytic parabolic induction of a locally algebraic character coincides with the universal unitary completion of the corresponding locally algebraic induction, provided that the character being induced satisfies a “non-critical slope” condition. (See Proposition 2.5 below.) In the second half of the paper we consider a certain unitary Banach space representation of GL2(Qp) obtained by p-adically completing the cohomology of classical modular curves. The mere existence of this representation implies that those locally algebraic parabolically induced representations of GL2(Qp) that arise from classical finite slope newforms have a non-trivial universal unitary completion (verifying a conjecture of Breuil for these representations), while applying Proposition 2.5 in this context enables us to give a new construction of p-adic L-functions attached to p-stabilized newforms of non-critical slope. Combining our construction with a representation theoretic definition of L-invariants due to Breuil [5], we are able to give a simple proof of the Mazur-Tate-Teitelbaum exceptional zero conjecture (in terms of Breuil’s definition of the L-invariant). The object of this note is two-fold. In its first half we consider the following problem in the representation theory of p-adic groups: When does a topologically irreducible continuous representation of a p-adic reductive group G on a p-adic topological vector space (perhaps satisfying some additional hypotheses) admit an embedding into a unitary representation of G on a p-adic Banach space? If G = GL2(Qp) and the representation considered is admissible locally algebraic, this problem has been raised by Breuil in connection with his ideas on a “p-adic Local Langlands correspondence for GL2(Qp)” [4, 5]. In the second half of the paper we explain a representation theoretic point of view on p-adic L-functions attached to modular forms. As an application, we prove a version of the Mazur-Tate-Teitelbaum exceptional zero conjecture [16, p. 46], using the representation theoretic definition of the L-invariant given by Breuil in [5] (which provides a reinterpretation of the definition given by Darmon [9] and Orton [18] in the weight two and higher weight cases respectively). The two halves of the paper are more closely related than they might appear at first glance. Indeed, the local tools discussed in the first half play a vital role in the The author would like to acknowledge the support of the National Science Foundation (award number DMS-0401545)