The Arithmetic Properties of Modular Forms and Hypergeometric Systems

Number theory is, essentially, the study of the properties of numbers. This seemingly simple concept leads to remarkably difficult unsolved problems in mathematics, with implications in areas such as biology, chemistry, computer science, and physics. This project focuses on investigating the connection between two fundamental objects in number theory: modular forms and hypergeometric functions. The theory of classical modular forms has long played an important role in number theory and was essential in Wiles’ proof of Fermat’s Last Theorem. More recently, generalized modular forms have become central objects of study, and can be understood through differential equations satisfied by classical modular forms. Special functions known as classical hypergeometric functions are known to satisfy very similar differential equations, suggesting a connection between hypergeometric functions and modular forms. In turn, hypergeometric functions provide arithmetic information for various mathematical objects, including multi-parameter families of Calabi-Yau manifolds leading to applications in string theory. An overall expectation is that hypergeometric functions provide a new direction in understanding the phenomena arising in mirror symmetry, one of the central research themes binding string theory and algebraic geometry. This project will make use of this connection to hypergeometric functions to study the arithmetic properties and applications of general modular forms. The broader impacts of this project include mentoring graduate and undergraduate students in research, organizing conferences and workshops, continuing to work on outreach programs with middle and high school students, and disseminating data and expository notes. Specifically, this project will study the properties of modular forms in relation to character sums and differential equations – especially those of hypergeometric type – using methods from arithmetic geometry, Galois theory, and Galois representations. The PI will focus on the exploration of modular forms on Shimura curves – including classical modular curves – which are moduli spaces of certain varieties. The main goals are: (1) to discover the arithmetic properties of modular forms on Shimura curves through explicit constructions; (2) to advance the understanding of hypergeometric systems and the modularity of hypergeometric Galois representations; and (3) to exploit the relations between hypergeometric functions and modular forms for arithmetic triangle groups to understand the fundamental properties of these two objects, such as their values at complex multiplication points and L-values. This project is jointly funded by the Algebra and Number Theory Program and the Established Program to Stimulate Competitive Research (EPSCoR). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.