Modular Forms for Noncongruence Subgroups

近年来，对同余子群模形式的研究取得了惊人的进展：谷山-志村-威尔猜想和塞尔猜想的解决，仅举几例。关于非同余子群的模形式，近年来也取得了较快的进展。虽然我们对非同余模形式的认识仍处于相对初级阶段，但这一主题已经显示出与几个富有成果的研究前沿的丰富联系：经典模形式、伽罗瓦表示、朗兰兹纲领和p进模形式。该提案的总体目标是继续由PI和她的合作者发展非同余模形式。本文主要研究三个目标：1)研究a . Scholl构造的非同同尖形上的伽罗瓦表示如何通过朗兰兹对应与经典自同构形式相关联，以及这种关系的应用。2)理解一个基本猜想，该猜想断言，如果所有系数都是代数的，则真正非同余模形式的傅里叶系数具有无界分母。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都对知识分子提出了挑战，而他们的思考反过来又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，以及教育学生。非同余模形式理论作为一个发展中的领域，包含了广泛的主题。一些合适的项目将被纳入研究生和高级本科生的学习和训练中。研究结果将通过参考期刊文章、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.