This proposal contains three projects, in number theory and its applications to combinatorics. The first project concerns the arithmetic of modular forms for noncongruence subgroups. For noncongruence subgroups whose associated modular curves have a model over the rationals, Atkin and Swinnerton-Dyer suggested very interesting congruence relations on the Fourier coefficients of cusp forms which are meant to replace the Hecke operators. On the other hand, Scholl has attached Galois representations to the space of noncongruence cusp forms. The ASD congruences together with the modularity of Scholl representations yield extremely interesting congruence relations between the Fourier coefficients of congruence and noncongruence forms. Some examples were constructed by the PI and her coauthors. Continuing her joint work with coauthors, the PI plans to apply the modularity lifting theorems to investigate when Scholl representations arise from Hilbert modular forms, and to use the modularity result and ASD congruence relations to establish the conjecture which says that the Fourier coefficients of genuinely algebraic noncongruence forms have unbounded denominators. The second project is to construct zeta functions of complexes arising from finite quotients of the Bruhat-Tits buildings. Such zeta function should be a rational function which encodes topological and spectral information of the complex, and which satisfies the Riemann Hypothesis if and only if the complex is Ramanujan. When dimension is one, this is the Ihara zeta function attached to a graph. In a very recent work, the PI and a PhD student did the 2-dimensional case. The PI proposes to explore the general case. The third project concerns low-density parity-check (LDPC) codes. The LDPC codes are equipped with very efficient decoding algorithms, which make them highly desirable in real world applications. The source of decoding errors is the pseudo-codewords. One way to understand these pseudo-codewords is to construct a suitable infinite series, called a zeta function, with each term corresponding to a pseudo-codeword. Such zeta function should be a rational function with good combinatorial property. This was done indirectly in a joint paper of the PI. The PI proposes to pursue a more direct approach.
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 combinatorics and coding theory, especially to solve real world problems. The study of interplay between these areas has turned out to be quite fruitful. This proposal is a continuation of the PI's effort to pursue the same general theme. Part of the research will be carried out by the PI's Ph D students. The results from this proposal will be disseminated broadly through the talks given by the PI in seminars, colloquia, conferences, short courses, and workshops. They will also be incorporated in the graduate courses taught by the PI. Weekly informal seminars will be conducted to integrate research with education and teaching. A conference is planned in 2010 to disseminate results from this project.
