West End Number Theory Seminar: Cameron Franc

The Shimura-Maass derivative is a differential operator which maps modular functions of weight $k$ to modular functions of weight $k+2$, but which does not preserve holomorphy. Shimura was the first to notice that, modulo powers of a CM-period, algebraic modular forms and their Shimura-Maass derivatives take algebraic values at CM-points. In this talk we will define a $p$-adic analogue of the Shimura-Maass operator. We will then define the ring of nearly rigid analytic modular forms as a subring of the continuous $\CC_p$-valued functions on the unramified points of the $p$-adic upper half plane; it is the smallest such ring containing the rigid analytic modular forms and which is closed uner the rigid Shimura-Maass operator. We will conclude with a rigid analogue of Shimura's classical algebraicity result for CM-values.