The Arithmetic of Hypergeometric Varieties and Noncongruence Modular Forms

点击翻译按钮获取中文摘要

Number theory is one of the scientific backbones of modern digital technology and has important applications in coding theory and cryptography. Beginning with the proof of Fermat's Last Theorem in the 1990's, number theory entered a new era, incorporating rich ideas and tools from other subjects, embracing rapidly-developing mathematical software packages and databases, and building multi-layer platforms and communities for collaborations. This research project investigates modular forms, which are central objects in number theory that occur throughout mathematics and physics. The work aims to broaden and deepen understanding in this fundamental area. The investigator plans to widen the impacts of the research program by continued mentoring of junior mathematicians and organization of national and international workshops and conferences. This project investigates the arithmetic of hypergeometric varieties and noncongruence modular forms. A majority of modular forms are noncongruence in the sense that their symmetry groups cannot be described in terms of congruences. The link between the two topics of study is manifest by the fact that all algebraic curves defined over number fields can be realized as modular curves of finite index subgroups of the modular group. Modular forms are arithmetic invariants of the modular curves. Among all algebraic curves, the generalized Legendre curves are particularly amiable models for both theoretical and computational investigations. Hypergeometric varieties are higher dimensional analogues of the generalized Legendre curves. The objectives of this research project are: (1) to translate a class of classical hypergeometric formulas to the finite field setting; (2) to use these transformation formulas to compute arithmetic invariants of the hypergeometric varieties; (3) to obtain new automorphy results for Galois representations associated with noncongruence modular forms based on explicit Galois representations arising from hypergeometric varieties; and (4) to further investigate other properties of noncongruence modular forms, including a longstanding unbounded denominator conjecture.