Joint IAS/Princeton University Number Theory Seminar

We introduce a new $p$-adic Maass-Shimura operator acting on a space of "generalized $p$-adic modular forms" (extending Katz's notion of $p$-adic modular forms) defined on the $p$-adic (preperfectoid) universal cover of Shimura curves. Using this operator, we construct new $p$-adic $L$-functions in the style of Katz, Bertolini-Darmon-Prasanna and Liu-Zhang-Zhang for Rankin-Selberg families over imaginary quadratic fields $K$ in the case where $p$ is inert or ramified in $K$. We also establish new $p$-adic Waldspurger formulas, relating $p$-adic logarithms of elliptic units and Heegner points to special values of these $p$-adic $L$-functions.