Theta Cycles of Modular Forms Modulo p 2

The theta cycle of a modular form modulo a prime p 5 is well understood.By contrast, the theta cycle modulo a power of p is still mysterious and experimentally erratic.Here we completely determine the theta cycle of a weight k < p modular form modulo p 2 on the initial segment of length p and we prove exact values or nontrivial bounds for the weight filtrations on p -2 further segments of length pk + 1.In particular, asymptotically as p - we establish 50% of the theta cycle exactly, and we provide nontrivial bounds for 100% of it.We determine the first two low points exactly and p-k+1 2further low points at regular positions.Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo p, and which disturb the otherwise regular structure in the segments that we exhibit.theta operator theta cycle low points MSC Primary: 11F33 MSC Secondary: 11F11 T HROUGHOUT, let p 5 be a prime.Given a quasi-modular form f on SL 2 (Z) with p-integral rational coefficients and an integer m 1, let p m ( f ) be the weight filtration of f modulo p m as defined in (1.1).Iteration of the theta operator (defined in (1.7)) yields the extended theta cycle of f modulo p m : p m f := p m f , p m 1 f , . . ., p m (p-1)p m-1 +m-1 f ,