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

点击翻译按钮获取中文摘要

"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.