Combinatorics and Number Theory V

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This proposal aims to investigate problems from two diverse areas in number theory andcombinatorics.The first project concerns arithmetic properties of modular forms onnoncongruence subgroups. To these forms Scholl attached motivic Galoisrepresentations, which are expected to be connected to automorphic forms according to Langlands philosophy. These Galois representations, unliketheir 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) automorphyof Scholl representations with special symmetries, exploit consequencesof automorphy, and study congruence properties ofFourier 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 early1970'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-dimensionalcomplexes 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 recentlywhen PI and her students obtained closed form expressions for zeta functions ofcomplexes 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 fundamentalresearch in number theory and to seek applications of number theoryto 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 ofthe 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 disseminatedbroadly through the talks given by the PI in seminars, colloquia,conferences, short courses, and workshops. They will also beincorporated in the graduate courses to be offered by the PI. Weeklyinformal seminars will be conducted to integrate research witheducation and teaching. The PI also plans to co-organize a conference in 2013 at Banffto disseminate results related to this proposal obtained by her and her students.