Modular Forms for Noncongruence Subgroups

近年来，同余子群的模形式的研究取得了惊人的进展：Taniyama-Shimura-Weil猜想和Serre猜想的解决，仅举几例。近年来，关于非同余子群的模形式的研究也取得了很大进展。虽然我们对非同余模形式的认识还处于起步阶段，但这门学科已经显示出它与几个卓有成效的研究前沿的丰富联系：经典模形式、伽罗瓦表示、朗兰兹规划和p-进模形式。这项建议的总体目标是继续由PI和她的合作者开发非一致模形式。该建议包含3个目标：1)研究A.Scholl构造的非同余尖点形式上的Galois表示与经典的自同构形式通过朗兰兹对应关系的联系，以及关系的应用。2)理解一个基本猜想，即如果所有系数都是代数的，则真非同余模形式的傅立叶系数有无界分母。3)探索非同余模形式的p-进性质及其在其他研究领域的应用。从希腊时代开始，数论中的许多重大问题挑战了人们的思维，他们的考虑反过来也提供了许多有用的应用。这一建议强调理论的发展，广泛的应用，以及对学生的教育。作为一个发展中的领域，非同余模型理论包含了广泛的主题。一些合适的项目将纳入研究生和高级本科生的学习和培训。结果将通过参考的期刊文章、研讨会和会议演讲以及研究生课程的主题广泛传播。

In recent years, stunning advances have been made in the study of modular forms for congruence subgroups: the settlements of the Taniyama-Shimura-Weil conjecture and Serre's conjecture, to name a few. On modular forms for noncongruence subgroups, rapid progress has also been made recently. While our knowledge of noncongruence modular forms is still in its relative infancy, the subject has already shown its rich connections with several fruitful research frontiers: classical modular forms, Galois representations, the Langlands program, and p-adic modular forms. The general aim of this proposal is to continue the development of noncongruence modular forms by the PI and her collaborators. The proposal contains 3 objectives: 1) To study when Galois representations attached to noncongruence cusp forms constructed by A. Scholl are related to classical automorphic forms via Langlands correspondence, as well as the applications of a relation. 2) To understand a fundamental conjecture which asserts that Fourier coefficients of genuine noncongruence modular forms have unbounded denominators if all coefficients are algebraic. 3) To explore p-adic properties of noncongruence modular forms and their applications to other research areas.Starting from the time of the Greeks, many great problems in number theory have challenged intellectual minds, and their considerations have provided numerous useful applications in turn. This proposal emphasizes theoretic developments, broad applications, as well as educating students. As a developing area, the theory of noncongruence modular form contains a wide spectrum of topics. Some suitable projects will be incorporated in the learning and training of graduate and advanced undergraduate students. The outcome will be disseminated widely through referred journal articles, seminar and conference talks, as well as topics for graduate courses.