Combinatorics and Number Theory IV

点击翻译按钮获取中文摘要

This proposal contains three projects, in number theory and itsapplications to combinatorics. The first project concerns thearithmetic of modular forms for noncongruence subgroups. Fornoncongruence subgroups whose associated modular curves have a modelover the rationals, Atkin and Swinnerton-Dyer suggested veryinteresting congruence relations on the Fourier coefficients of cuspforms which are meant to replace the Hecke operators. On the otherhand, Scholl has attached Galois representations to the space ofnoncongruence cusp forms. The ASD congruences together with themodularity of Scholl representations yield extremely interestingcongruence relations between the Fourier coefficients of congruenceand noncongruence forms. Some examples were constructed by the PIand her coauthors. Continuing her joint work with coauthors, the PIplans to apply the modularity lifting theorems to investigate whenScholl representations arise from Hilbert modular forms, and to usethe modularity result and ASD congruence relations to establish theconjecture which says that the Fourier coefficients of genuinelyalgebraic noncongruence forms have unbounded denominators. Thesecond project is to construct zeta functions of complexes arisingfrom finite quotients of the Bruhat-Tits buildings. Such zetafunction should be a rational function which encodes topological andspectral information of the complex, and which satisfies the RiemannHypothesis if and only if the complex is Ramanujan. When dimensionis one, this is the Ihara zeta function attached to a graph. In avery recent work, the PI and a PhD student did the 2-dimensionalcase. The PI proposes to explore the general case. The third projectconcerns low-density parity-check (LDPC) codes. The LDPC codes areequipped with very efficient decoding algorithms, which make themhighly desirable in real world applications. The source of decodingerrors is the pseudo-codewords. One way to understand thesepseudo-codewords is to construct a suitable infinite series, calleda zeta function, with each term corresponding to a pseudo-codeword.Such zeta function should be a rational function with goodcombinatorial property. This was done indirectly in a joint paper ofthe PI. The PI proposes to pursue a more direct approach.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 coding theory, especially to solve real worldproblems. The study of interplay between these areas has turned outto 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 Dstudents. 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 taught by the PI. Weeklyinformal seminars will be conducted to integrate research witheducation and teaching. A conference is planned in 2010 todisseminate results from this project.