Differential operators on modular forms, and Galois representations
Since a modular form is a holomorphic function, it is tempting to take its derivative. However, this destroys the modularity property. Several approaches exist for "fixing" this problem, and the resulting objects have many arithmetic applications.
I will discuss such differential operators on various types of modular forms (mod p), indicate a few ways of constructing them, and describe the effect of these operators on the Galois representations attached to Hecke eigenforms.
(This is an amalgamation of various projects joint with Owen Colman, Ellen Eischen, Max Flander, Elena Mantovan, Angus McAndrew, and Takuya Yamauchi.)
Alex Ghitza, University of Melbourne