Galois Representations and Modular Forms

[00:70659]申请人将在代数数论的两个基本问题的背景下研究伽罗瓦表示和模形式之间的联系。第一个问题是确定哪些伽罗瓦表示与自同构表示相关联。第二个问题是将变量或动机的l函数的特殊值与相关的代数量（如Selmer群的阶）联系起来。在第一个问题的方向上，提议者打算对他最近关于二维p进伽罗瓦表示的各种族的模块化的工作进行扩展。特别地，提议者打算探索对更高维度表示的可能扩展。在第二个问题的方向上，作者将研究高阶群（通常涉及l函数的特殊值）上的爱森斯坦级数常数项的可整除性质对这些爱森斯坦级数的Hecke特征值与尖形的Hecke特征值之间的同余的影响。这两个问题都属于“算术几何”的范畴，算术几何是数学的一个分支，它试图将复杂的数学应用于数论中通常易于陈述但难以解决的问题。算术几何最容易解决的问题包括：1)计算多项式系统的整数解的数量；2)理解l函数（附加在多项式系统上的函数，并编码有关其解的信息）。这些问题在应用中越来越重要。计算、密码学和编码理论的最新进展依赖于这些问题的解决方案，这些领域的许多进展也是如此。

0070659The proposer will investigate connections between Galois representations and modular forms in the context of two fundamental problems in algebraic number theory. The first of these problems is to determine which Galois representations are associated to automorphic representations. The second problem is to relate special values of L-functions of varieties or motives to associated algebraic quantities (such as orders of Selmer groups). In the direction of the first problem the proposer intends to pursue extensions of his recent work on the modularity of various families of two-dimensional p-adic Galois representations. In particular, the proposer intends to explore possible extensions to representations of higher dimension. In the direction of the second problem, the proposer will investigate the influence of divisibility properties of constant terms of Eisenstein series on higher-rank groups (which often involve special values of L-functions) on congruences between the Hecke eigenvalues of these Eisenstein series and those of cusp forms.Both of these problems fall under the rubric "Arithmetic Geometry," a branch of mathematics which attempts to apply sophisticated mathematics to the often easy-to-state, but-hard-to-solve problems of number theory. The problems most readily tackled by arithmetic geometry include 1) counting the number of integer solutions to systems of polynomials and 2) understanding L-functions (functions attached to systems of polynomials and which encode information about their solutions). These problems are of increasing importance in applications. Recent advances in computing, cryptography, and coding theory have depended on solutions to these problems as do many proposed advances in these areas.