New Directions in the Theory of Automorphic Forms

这个研究项目涉及数学最古老的分支--数论。更具体地说，它侧重于自守形式的研究。自守形式是一类非常特殊的函数，它形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。自守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。经典上，自守形式的特殊值为不位于真实的直线上的有理数的扩展提供了克罗内克问题的解决方案。最近，自守形式在费马大定理的证明中发挥了关键作用。本项目旨在进一步探索自守形式与其他数学结构的联系。拟议研究的主要目标是进一步理解和开发自守形式与二次数域之间的关系。这种关系是非常丰富和团结的研究不同的对象，如Heegner点，封闭测地线，双曲拉普拉斯，Kloosterman和，L-函数和类领域。虚二次域的理论一般来说发展得更好，也更简单，特别是与类域理论有关。研究者将主要集中在真实的二次情形。在一个方向上，他们将研究一些新的几何不变量相关的真实的二次域，最近推出。这些不变量是由模闭测地线限定的曲面。调查人员将研究分布的面积的表面，特别是因为这涉及到理想类，也调查了一些新的几何问题有关的封闭测地线。他们计划用自守形式的傅里叶系数来表示真实的二次域的各种不变量（例如模函数的曲面积分）。这自然导致的问题，涉及款项Kloosterman和扩展最近的工作，统一估计等款项。

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.