[论文解读] Differential Operators and Families of Automorphic Forms on Unitary Groups of Arbitrary Signature

Ellen Eischen, Jessica Fintzen+2 more|arXiv (Cornell University)|Nov 20, 2015

Advanced Algebra and Geometry参考文献 20被引用 9

一句话总结

本文开发了p进微分算子，并在任意亏格的酉群上构造了自守形式的p进族，通过使用Serre–Tate展开式克服了q-展开式缺失的问题。它构造了一个取值于p进自守形式的p进测度，并证明了该测度对Eisenstein级数的插值性质，将先前工作推广至非(n,n)亏格情形，从而为p进L-函数的新构造提供了可能。

ABSTRACT

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 $(a, b)$, $a eq b$. 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.

研究动机与目标

• 将Serre与Katz利用同余关系与微分算子构造p进自守形式族的策略，推广至非(n,n)亏格的情形。
• 通过引入Serre–Tate展开式作为替代，克服酉群在非(n,n)亏格下q-展开式缺失的问题。
• 构造一个取值于p进自守形式的p进测度，使其对任意亏格的酉群上的Eisenstein级数实现插值。
• 为p进L-函数的构造提供基础工具，如[EHLS16]中最近完成的工作所示。
• 通过Serre–Tate坐标与微分算子，将p进q-展开式原理推广至酉群。

提出的方法

• 在酉群上的自守形式中，使用Serre–Tate展开式（在普通CM点处具有自然坐标的展开式）代替q-展开式。
• 通过Gauss–Manin联络与Kodaira–Spencer同态，分析p进微分算子在Serre–Tate展开式上的作用。
• 应用一种新颖的提升论证，构造一个取值于p进自守形式的p进测度，定义在Xp × T(Zp)上。
• 依赖于适配至Serre–Tate展开式的p进q-展开式原理，利用抽象Kummer同余式确保整数性与插值性。
• 将微分算子的像投影至不可约最高权表示，以定义自守形式的插值。
• 通过Hecke特征标与p进特征标，建立测度积分与缩放后的Eisenstein级数之间的对应关系。

实验结果

研究问题

• RQ1 当q-展开式缺失时，如何在酉群上的自守形式上定义并计算p进微分算子？
• RQ2 能否为任意亏格(a,b)（其中a ≠ b）的酉群构造同余关系与p进自守形式族？
• RQ3 Serre–Tate展开式在替代q-展开式以实现插值与微分算子作用中起什么作用？
• RQ4 如何构造一个p进测度，使其在任意亏格下对p进族中的Eisenstein级数实现插值？
• RQ5 p进q-展开式原理在多大程度上可通过Serre–Tate坐标与微分算子推广至酉群？

主要发现

• 本文明确描述了p进微分算子在任意亏格酉群上自守形式的Serre–Tate展开式上的作用。
• 它在Xp × T(Zp)上构造了一个取值于G′上p进自守形式空间的p进测度µG′，满足∫Xp×T(Zp) ˜χψκµG′ = resΘκGk,ν,Fχu,ψ。
• 对于纯H′-对称权κ，测度积分精确恢复Eisenstein级数Gk,ν,Fχu,ψ,κ。
• 该构造将先前关于(n,n)亏格p进族的结果推广至任意亏格，消除了以往的限制。
• 该方法使得通过微分算子与最高权分量的投影，实现自守形式在普通CM点处的p进插值（至周期为止）。
• 该工作为[EHLS16]中最近完成的酉群p进L-函数的构造提供了关键技术基础。