This proposal aims to investigate problems from two diverse areas in number theory and combinatorics. The first project concerns arithmetic properties of modular forms on noncongruence subgroups. To these forms Scholl attached motivic Galois representations, which are expected to be connected to automorphic forms according to Langlands philosophy. These Galois representations, unlike their classical counterpart, cannot be broken into pieces in general, and should be related to automorphic forms on symplectic and orthogonal groups. Building upon her past work, PI will investigate the (potential) automorphy of Scholl representations with special symmetries, exploit consequences of automorphy, and study congruence properties of Fourier coefficients of noncongruence forms as proposed by Atkin and Swinnerton-Dyer. The second project is to study the interplay between combinatorics, group theory and number theory through associated zeta functions. Zeta functions of varieties defined over finite fields are well-understood. Their most well-known properties are described by Weil conjectures, established in early 1970's. Finite simplicial complexes are combinatorial analog of such varieties. They are expected to have zeta functions enjoying similar properties, except that Riemann Hypothesis will hold only for complexes which are spectrally optimal. One-dimensional complexes are graphs, whose zeta functions have been studied since the work of Ihara in 1966. The zeta functions for higher dimensional complexes became known only recently when PI and her students obtained closed form expressions for zeta functions of complexes arising as quotients of the buildings of certain rank-2 Chevalley groups over p-adic fields. The approaches are mostly representation-theoretic. The PI proposes to find zeta identities for complexes arising from other groups. She also intends to explore combinatorial interpretations of these identities using the Selberg trace formula, with an eye towards establishing a connection between complex zeta functions and automorphic forms.
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 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 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 to be offered by the PI. Weekly informal seminars will be conducted to integrate research with education and teaching. The PI also plans to co-organize a conference in 2013 at Banff to disseminate results related to this proposal obtained by her and her students.
