Automorphic Forms, L-functions and Galois Representations

作者提出：(1)利用l -函数、迹公式、同余、Hasse不变形式的限制和伪表示，将GL(4)/F上的某些关联形式转移到合适的酉群上，从而将伽罗什表示附加到任意CM域k（具有全实数子域F）上权k 1的尖形上；（ii）构造GSp(4)/Q上的某些特殊全纯形式，研究它们对GL(4)/Q的提升，并推导出某些伽罗瓦表示的结果；（iii）继续与D. Prasad合作，改进GL(n)的自对偶表示的局部朗兰兹对应。自同构形式是自同构形式的研究领域。最简单但又不那么简单的基本问题示例如下：从一系列数字{a_0， a_1, a_2, a_3，…， a_n，…}并考虑“生成函数”f(q) = a_0 + a_1q + a_2q^2 +…+ a_nq^n + ....，其中q是哑变量。一个基本问题是知道f(q)何时满足“隐藏对称”。为了详细说明，写q = exp(2\pi iz)，其中z是正虚部的复数，并设q* = exp（-2\pi i/z）。人们经常寻找的是一对(f（q), f(q*)）之间的关系，这在弦物理学和组合学等不同的领域也会出现。一种是f(q*) =(-log q/2\ i)^k f(q) f的权值为k。这种对称性的存在意味着我们开始的序列{a_n}具有不可思议的性质。例如，当a_n是乘法时，即当a_{mn} = a_ma_n时，n相对素数，a_0=0, a_1=1，则有一个来自几何的相关二维伽罗瓦表示R，其相关的“l函数”等于1 + a_2/2^s + a_3/3^s +…，意味着对于每一个素数p， a_p =u_p + 1/u_p，其中u_p是绝对值p^{(k-1)/2}的代数整数；其中|a_p|以2p^{(k-1)/2}为界，不能用先验分析估计证明。要记住的一个关键例子是无处不在的函数q{(1-q)（1-q^2)（1-q^3）…）}^{24} = q+tau_2q^2 + tau_3q^3 +…，它的质量是12。一般的“朗兰兹纲领”设想了许多这样的情况，它们涉及到一系列对称（“模块化”），这些对称很难明确地写下来，但由于其深远的影响，它们仍然是非常重要的。例如，怀尔斯著名的费马大定理证明的一个关键成分就利用了这个程序中得到的一个结果。在他目前（即将成为以前的）NSF提案的相关工作中，P.I.证明了给定两个函数f(q), g(q)分别附加于{a_n}， {b_n}上，承认某些权值的隐对称性，乘积序列{a_nb_n}与一个4次模对象相关联。这有以下后果。假设f和g的权值相同，进一步假设对于几乎所有素数p， a_p^2都等于b_p^2，那么f等于g。

AUTOMORPHIC FORMS, L-FUNCTIONS AND GALOIS REPRESENTATIONS The princial investigator proposes to do the following: (i) attachGalois representations to cusp forms of weight k 1 over any CM field K (with totally real subfield F) by transferring certain associated forms on GL(4)/F to suitable unitary groups by making use of L-functions, trace formula, congruences, restrictions of Hasse invariant forms and pseudo-representations; (ii) construct certain special holomorphic forms on GSp(4)/Q, study their lifting to GL(4)/Q, and derive consequences for certain Galois representations; and (iii) to continue ongoing work with D. Prasad on a refinement of the local Langlands correspondence for self-dual representations of GL(n). The field of research of the P.I. is Automorphic Forms. The simplest, yet not so simple, instance of the basic problem of the field is the following: Start with a sequence of numbers {a_0, a_1 , a_2, a_3, .., a_n, ...} and consider the "generating function" f(q) = a_0 + a_1q + a_2q^2 + ... + a_nq^n + ...., where q is a dummy variable. A fundamental question is to know when f(q) satisfies a "hidden symmetry". To elaborate, write q = exp(2\pi iz), with z a complex number of positive imaginary part, and set q* = exp(-2\pi i/z). What one is often looking for, and this shows up in disparate fields like string Physics and combinatorics, is a relationship between the pair (f(q), f(q*)). One says that f has weight k if f(q*) =(-log q/2\pi i)^k f(q). The existence of such a symmetry implies that the sequence {a_n} we started with has miraculous properties. For example, when the a_n are multiplicative, i.e., when a_{mn} = a_ma_n for m,n relatively prime, with a_0=0 and a_1=1, then there is an associated 2-dimensional Galois representation R coming from geometry whose associated "L-function" equals 1 + a_2/2^s + a_3/3^s + ..., implying that for each prime p, a_p =u_p + 1/u_p with u_p an algebraic integer of absolute value p^{(k-1)/2}; in particular, |a_p| is bounded by 2p^{(k-1)/2}, which is not provable by an apriori analytic estimate. A key example to keep in mind is the ubiquitous Delta function q{(1-q)(1-q^2)(1-q^3)...)}^{24} = q+tau_2q^2 + tau_3q^3 + ..., which has weight 12. The general "Langlands program" envisions many such occurrances, and they involve a family of symmetries ("modularity") which are complicated to write down explicitly, but are nevertheless very important to pursue due to their far-reaching consequences. For example, one key ingredient of the celebrated proof of Fermat's last theorem by Wiles makes use of a result obtained in this program. In his work related to hiscurrent (about to become preious) NSF proposal, the P.I. proved that given two functions f(q), g(q) as above attached to {a_n}, {b_n} respectively, admitting hidden symmetries of some weights, the product sequence {a_nb_n} is associated to a modular object of degree 4. This has the following consequence. Suppose f, g have the same weights, and suppose further that a_p^2 equals b_p^2 for almost all primes p. Then f equals g.