Modular forms are often studied for the interesting arithmetic properties of their Fourier coefficients. We can reduce these Fourier coefficients modulo a prime power and find congruences between modular forms. In this talk, I will discuss the theta cycle of a modular form f, which is a list of the lowest weights such that repeated derivatives of f are congruent to a modular form of that weight. The mod p theta cycle is well-understood, but the theta cycle for higher prime powers is still very mysterious. I will discuss recent work of Ahlgren, Raum, and Richter where they determined portions of the mod p^2 theta cycle, and I will present my new work joint with these authors where we have now determined approximately 50% of the mod p^2 theta cycle (when the starting weight is < p).
Talk by Amy Woodall.