New Directions in the Theory of Automorphic Forms

这项研究项目涉及数论这一最古老的数学分支。更具体地说，它侧重于研究自同构形式。自同构型是一类非常特殊的函数，它们构成了连接代数数论的离散对象和解析数论的连续对象的重要桥梁。自同构形式被用作研究数论函数的工具，例如，通过测量它们的增长率，发现它们的公式，或证明它们满足的关系。经典的自同构形式的特殊值为不在实线上的有理数的扩张提供了Kronecker问题的解决方案。最近，自同构形式在费马大定理的证明中发挥了关键作用。该项目旨在进一步探索自同构形式与其他数学结构之间的联系，其主要研究目的是进一步理解和开发自同构形式与二次数域之间的关系。这种关系非常丰富，并将对不同对象的研究统一在一起，如Heegner点、闭测地线、双曲拉普拉斯、Klosterman和、L函数和类域。虚二次域的理论总体上是比较发达和简单的，特别是在与类域理论相关的方面。调查人员将主要集中在真正的二次案件上。在一个方向上，他们将研究一些新的几何不变量，这些几何不变量与最近引入的实二次场有关。这些不变量是由模闭测地线限定的某些曲面。研究人员将研究曲面的面积分布，特别是当这与理想类有关时，还将研究一些关于闭测地线的新几何问题。他们计划用自同构形式的傅里叶系数来表示实二次域的各种不变量(如模函数的曲面积分)。这自然会导致涉及Krousterman和的问题，并导致最近关于这类和的统一估计的工作的扩展。

This research project concerns number theory, the oldest branch of mathematics. More specifically, it focuses on the study of automorphic forms. Automorphic forms are a very special class of functions that form an important bridge connecting the discrete objects of algebraic number theory and the continuous objects of analytic number theory. Automorphic forms are used as tools to study number theoretic functions, e.g. by measuring their rates of growth, discovering formulas for them, or proving relations that they satisfy. Classically, special values of automorphic forms provided the solution to Kronecker's problem for extensions of the rational numbers that do not lie in the real line. More recently, automorphic forms played a pivotal role in the proof of Fermat's Last Theorem. This project aims to further explore the connection of automorphic forms to other mathematical structures.The main object of the proposed research is to further understand and exploit the relationship between automorphic forms and quadratic number fields. This relationship is exceedingly rich and unites the study of diverse objects such as Heegner points, closed geodesics, the hyperbolic Laplacian, Kloosterman sums, L-functions and class fields. The theory for imaginary quadratic fields is in general better developed and simpler, especially in relation to class field theory. The investigators will concentrate mostly on the real quadratic case. In one direction, they will study some new geometric invariants associated to real quadratic fields that were introduced recently. These invariants are certain surfaces that are bounded by modular closed geodesics. The investigators will study the distribution of the areas of the surfaces, especially as this relates to ideal classes and also investigate some new geometric problems about the closed geodesics. They plan to express various invariants for real quadratic fields (such as surface integrals of modular functions) in terms of the Fourier coefficients of automorphic forms. This naturally leads to problems involving sums of Kloosterman sums and to extensions of recent work on uniform estimates for such sums.