We survey some recent developments in the theory of vector-valued modular forms for SL2(ℤ), focusing especially on our recent and ongoing joint work with Frank Calegari and Yunqing Tang that proved the Unbounded Denominators conjecture as one application.
The first talk will be an introduction to noncongruence modular forms, from one side, and from another side to arithmetic algebraization methods. We will discuss how to connect these two subjects, and the kind of further applications that arithmetic algebraization methods may have to offer in number theory. After the basic examples and some history, we will turn to Bost’s slopes method of Arakelov theory for the technical underpinning of our proofs.
In the second talk, I will establish a new equivariant holonomy bound and apply it to prove the Unbounded Denominators conjecture of Atkin, Swinnerton-Dyer, and Mason. This will be a new argument alternative to our original proof in (F. Calegari, V. Dimitrov, Y. Tang: The unbounded denominators conjecture).
This is the first part of two talks, the second of which may be found here.
This video is part of Harvard University‘s conference Current Developments in Mathematics 2023.
