Automorphic Forms, L-functions and Galois Representations

利用L函数、迹公式、同余、哈斯不变形式的限制和伪表示，将GL(4)/F上的某些伴随形式转移到合适的酉群上，从而在任意CM域K(具有全实子域F)上将权为k1的尖点形式附加Galois表示；(Ii)在GSP(4)/q上构造某些特殊的全纯形式，研究它们对GL(4)/q的提升，并得到某些Galois表示的结果；以及(Iii)继续与D.Prasad就GL(N)的自对偶表示的局部朗兰兹对应的精化进行工作。P.I.的研究领域是自同构形式。域的基本问题的最简单但不那么简单的例子如下：从一个数字序列{a_0，a_1，a_2，a_3，.，a_n，...}开始，考虑“生成函数”f(Q)=a_0+a_1q+a_2q^2+...+a_nq^n+...，其中q是一个伪变量。一个基本的问题是知道f(Q)何时满足“隐藏对称性”。为了详细说明，编写q=exp(2\pi iz)，其中z为正虚部的复数，并设置q*=exp(-2\pi i/z)。人们经常寻找的，这在弦物理和组合学等不同的领域中都可以看到，是这对(f(Q)，f(q*))之间的关系。如果f(q*)=(-logq/2\pi)^k f(Q)，则称f有权k。这种对称性的存在意味着我们开始的序列{a_n}具有神奇的性质。例如，当a_n是可乘的，即对m，n相对素数，a_0=0，a_1=1，a_n=a_ma_n时，则存在一个相关的二维伽罗瓦表示R，它来自几何，其相关的“L函数”等于1+a_2/2^S+a_3/3^S+…，这意味着对于每个素数p，a_p=u_p+1/u_p，u_p是绝对值为p^{(k-1)/2}的代数整数；特别地，|a_p|由2p^{(k-1)/2}有界，这不能用先验分析估计来证明。一个需要记住的关键例子是无处不在的Delta函数q{(1-q)(1-q^2)(1-q^3)...)}^{24}=q+tau_2q^2+tau_3q^3+...，它的权重为12。一般的“朗兰兹程序”设想了许多这样的情况，它们涉及一族对称性(“模性”)，显式地写下来很复杂，但由于它们的深远后果，追求它是非常重要的。例如，Wiles著名的费马大定理证明中的一个关键因素利用了在这个程序中获得的结果。在他关于当前(即将成为先验的)NSF提议的工作中，P.I.证明了给定两个函数f(Q)，g(Q)分别附加到{a_n}，{b_n}，并且允许某些权重的隐藏对称性，乘积序列{a_nb_n}与一个4次模对象相联系。假设f，g具有相同的权重，并且进一步假设对几乎所有素数p，a_p^2等于b_p^2，则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.