This proposal contains two projects, in number theory and its applications to combinatorics. The first project concerns modular forms for noncongruence groups and their connections to forms for congruence groups. The arithmetic of the forms for noncongruence groups is little understood. Based on their numerical data, Atkin and Swinnerton-Dyer (ASD) suggest very interesting congruence relations to be satisfied by a basis of cusp forms for a noncongruence group. A recent joint work by the PI, Long and Yang gives the first two-dimensional example establishing the ASD congruence relations between forms for noncongruence and congruence groups. To move forward, the PI plans to systematically study the structure of the cusp forms for noncongruence groups, to explore their relationship with forms for congruence groups, and to tackle the conjecture that algebraic cusp forms genuinely living on noncongruence groups are distinguished by their Fourier coefficients having unbounded denominators. The second is on the interplay between automorphic forms and Ramanujan hypergraphs, and connections between hypergraphs and LDPC codes. Ramanujan hypergraphs are higher dimensional analogue of Ramanujan graphs, which are known to have broad applications. The PI proposes to study whether the rich interplay between combinatorics and number theory, which exists for Ramanujan graphs, extends to Ramanujan hypergraphs. On the applied side, the PI plans to construct good LDPC codes using Ramanujan hypergraphs, and to generalize her joint work with Koetter, Vontobel, and Walker on characterizing pseudo-codewords for LDPC codes from attached to graphs to attached to hypergraphs.
It has been the PI's long term research goal to do fundamental research in number theory and to seek applications of number theory to graph theory and coding theory, especially to solve real world problems. The study of interplay between these areas has turned out to be quite fruitful. The PI has applied deep results in number theory to construct efficient communication networks; and conversely, investigations in graph theory inspired very interesting and unexpected results in number theory. This proposal is a continuation of the PI's effort to pursue the same general theme. The first project of the proposal lies in basic research, to understand the arithmetics of modular forms for noncongruence subgroups. The second project is to explore the interplay in arithmetic and connections between automorphic forms and Ramanujan hypergraphs, extending PI's previous success on Ramanujan graphs. A potential application is to LDPC codes. The very efficient encoding and decoding algorithms for LDPC codes together with their extensive applications make them very hot research topic in coding theory. A primary goal of the second project is to construct good LDPC codes using Ramanujan hypergraphs, and to understand the "wrongly" decoded words arising from the rapid decoding algorithm. A conference is planned in 2007 to disseminate the results of this proposal.
