Differential Operators and Families of Automorphic Forms on Unitary Groups of Arbitrary Signature
Differential Operators and Families of Automorphic Forms on Unitary Groups of Arbitrary Signature
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DOI：
10.25537/dm.2018v23.445-495
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发表时间：
2015-11
影响因子：
0.9
通讯作者：
E. Eischen;Jessica Fintzen;E. Mantovan;Ila Varma
中科院分区：
文献类型：
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作者：
E. Eischen;Jessica Fintzen;E. Mantovan;Ila Varma
在20世纪70年代，塞尔利用同余之间的q-展开系数的爱森斯坦系列生产p-adic家庭的爱森斯坦系列，反过来，p-adic zeta函数。部分通过与更近期的机器集成，包括Katz对p-adic微分算子的方法，他的策略影响了四十年的发展。以前的论文采用卡茨和塞尔的想法利用微分算子和同余产生家庭的自守形式依赖于自守形式的q-扩展至关重要。本文的首要目标是使该策略适用于酉群上的自守形式，当签名是（a，B），a ∈ B的形式时，该自守形式缺乏q-扩展。特别是，本文完全消除了以前的工作中存在的签名的限制。作为中间步骤，我们实现了两个关键目标。首先，部分地通过仔细分析Young对称化子对Serre-Tate展开的作用，我们明确地描述了微分算子对任意签名酉群上自守形式的Serre-Tate展开的作用。作为一个直接的后果，对于每个酉群，我们得到的同余和家庭类似的研究卡茨和塞尔。其次，通过一个新的提升参数，我们构造了一个p-adic测度，其取值于任何指定签名的酉群上的p-adic自守形式空间。我们将这一措施的价值观显式p-adic家庭的爱森斯坦系列。我们的结果的一个应用是最近完成的建设p-adic L-功能的酉群的第一个命名的作者，哈里斯，李，斯金纳。
In the 1970's, Serre exploited congruences between q-expansion coefficients of Eisenstein series to produce p-adic families of Eisenstein series and, in turn, p-adic zeta functions. Partly through integration with more recent machinery, including Katz's approach to p-adic differential operators, his strategy has influenced four decades of developments. Prior papers employing Katz's and Serre's ideas exploiting differential operators and congruences to produce families of automorphic forms rely crucially on q-expansions of automorphic forms. The overarching goal of the present paper is to adapt the strategy to automorphic forms on unitary groups, which lack q-expansions when the signature is of the form (a,b), a≠b. In particular, this paper completely removes the restrictions on the signature present in prior work. As intermediate steps, we achieve two key objectives. First, partly by carefully analyzing the action of the Young symmetrizer on Serre-Tate expansions, we explicitly describe the action of differential operators on the Serre-Tate expansions of automorphic forms on unitary groups of arbitrary signature. As a direct consequence, for each unitary group, we obtain congruences and families analogous to those studied by Katz and Serre. Second, via a novel lifting argument, we construct a p-adic measure taking values in the space of p-adic automorphic forms on unitary groups of any prescribed signature. We relate the values of this measure to an explicit p-adic family of Eisenstein series. One application of our results is to the recently completed construction of p-adic L-functions for unitary groups by the first named author, Harris, Li, and Skinner.