"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 deep and 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-standing open problems, including Wiles' proof of Fermat's Last Theorem. In this proposal, the investigator proposes to study the distribution of the Fourier coefficients of modular forms, with a focus on 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 of the oldest branches of mathematics. In 1770, Lagrange proved that every positive integer is a sum of four squares. This notable result has motivated a significant amount of present day research. A notable example is the recent work of Manjul Bhargava and Jonathan Hanke in which modular forms are used to classify positive-definite quadratic forms representing all positive integers. One application of the proposed research is a classification of positive-definite quadratic forms representing all odd integers.
