The distribution of the Fourier coefficients of modular forms and arithmetic applications

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。“世纪，（和21世纪初），在模形式、椭圆曲线、二次型、L-函数、伽罗瓦表示和零星有限单群的表示理论之间发现了许多深刻而令人惊讶的联系。这些联系导致了许多长期悬而未决的问题的解决，包括怀尔斯对费马大定理的证明。在这个提议中，研究者建议研究模形式的傅里叶系数的分布，重点是Hecke特征形的非平凡线性组合形式。其他主题包括算术动力学，非线性递归关系和模形式mod p。建议的研究是在数论领域，数学的最古老的分支之一。1770年，拉格朗日证明了每一个正整数都是四个平方的和。一个显著的例子是Manjul Bhargava和Jonathan Hanke最近的工作，其中使用模形式来分类表示所有正整数的正定二次型。所提出的研究的一个应用是正定quadraticforms表示所有奇数的分类。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."During the 20th century (and the beginning of the 21st), a number of deepand surprising connections have been found between modular forms,elliptic curves, quadratic forms, L-functions, Galois representations,and the representation theory of the sporadic finite simple groups.These connections have led to the resolution of a number of long-standingopen problems, including Wiles' proof of Fermat's Last Theorem.In this proposal, the investigator proposes to study thedistribution of the Fourier coefficients of modular forms, with a focuson forms that are non-trivial linear combinations of Hecke eigenforms.Other topics include arithmetic dynamics, non-linear recurrence relations,and modular forms mod p.The proposed research is in the area of number theory, one ofthe oldest branches of mathematics. In 1770, Lagrangeproved that every positive integer is a sum of four squares. Thisnotable result has motivated a significant amount of present day research.A notable example is the recent work of Manjul Bhargava andJonathan Hanke in which modular forms are used to classify positive-definitequadratic forms representing all positive integers. One applicationof the proposed research is a classification of positive-definite quadraticforms representing all odd integers.