Can you differentiate integers?

Presenter: Dr. Debargha Banerjee, Indian Institute of Science Education and Research

If we start with mod $p$ objects, it may or may not have lifts to characteristic zero objects. Buium introduced differential modular forms in a new geometry by using a close analogy with function field situation. In this new geometry, modular forms modulo $p$ always have lifting to characteristic zero modular forms. In this talk, we will introduce the theory of modular forms and differential modular forms. Differential modular forms are obtained by differentiating modular forms in this new way. In more fancy language, these are the modular forms obtained by applying the arithmetic jet space functors (adjoint to the Witt vector functors) to the ring of modular forms. We show that these differential modular forms help us to detect the ordinary elliptic curves and elliptic curves with Frobenius lifts. The whole process is executed by a new analogue of differentials applied on integers.